PART VIII.--PHYSIOGRAPHICAL GEOLOGY

This department of geological inquiry investigates the origin and history of the present topographical features of the land. As these features must obviously be related to those of earlier time which are recorded in the rocks of the earth's crust, they cannot be satisfactorily studied until at least the main outlines of the history of these rocks have been traced. Hence physiographical research comes appropriately after the other branches of the science have been considered.

From the stratigraphy of the terrestrial crust we learn that by far the largest part of the area of dry land is built up of marine formations; and therefore that the present land is not an aboriginal portion of the earth's surface, but has been overspread by the sea in which its rocks were mainly accumulated. We further discover that this submergence of the land did not happen once only, but again and again in past ages and in all parts of the world. Yet although the terrestrial areas varied much from age to age in their extent and in their distribution, being at one time more continental, at another more insular, there is reason to believe that these successive diminutions and expansions have on the whole been effected within, or not far outside, the limits of the existing continents. There is no evidence that any portion of the present land ever lay under the deeper parts of the ocean. The abysmal deposits of the ocean-floor have no true representatives among the sedimentary formations anywhere visible on the land. Nor, on the other hand, can it be shown that any part of the existing ocean abysses ever rose above sea-level into dry land. Hence geologists have drawn the inference that the ocean basins have probably been always where they now are; and that although the continental areas have often been narrowed by submergence and by denudation, there has probably seldom or never been a complete disappearance of land. The fact that the sedimentary formations of each successive geological period consist to so large an extent of mechanically formed terrigenous detritus, affords good evidence of the coexistence of tracts of land as well as of extensive denudation.

_The Geological Record or Order of Succession of the Stratified Formations of the Earth's Crust._

+---+---+-------------------------------------------+----------------------------------+ | | | Europe. | North America. | +---+---+-------------------------------------------+----------------------------------+ | Q | \ Historic, up to the present time. | Similar to the European | | u | \ Prehistoric, comprising deposits of | development, but with scantier | | a | \ the Iron, Bronze, and later | traces of the presence of man. | | t | \ Stone Ages. | | | e | \ Neolithic--alluvium, peat, lake- | | | r | Recent, \ dwellings, loess, &c. | | | n | Post- | Palaeolithic--river-gravels, cave- | | | a | glacial | deposits, &c. | | | r | or | | | | y | Human. | | | | | | | | | o | | | | | r +---------+-------------------------------------+----------------------------------+ | | Pleist- | Older Loess and valley-gravels; | As in Europe, it is hardly | | P | ocene | cave-deposits. | possible to assign a definite | | o | or | Strand-lines or raised beaches; | chronological place to each of | | s | Glacial.| youngest moraines. | the various deposits of this | | t | | Upper Boulder-clays; eskers; marine | period, terrestrial and marine.| | | | | sands and clays. | They generally resemble the | | T | | Interglacial deposits. | European series. The | | e | | Lower boulder-clay or Till, with | characteristic marine, | | r | | striated rock-surfaces below. | fluviatile and lacustrine | | t | | | terraces, which overlie the | | i | / | older drifts, have been | | a | / | classed as the Champlain Group.| | r | / | | | y | / | | | . | / | | +---+---+-------------------------------------------+----------------------------------+ | | P | Newer:--English Forest-Bed Group; Red and | On the Atlantic border | | | l | Norwich Crag; Amstelian and Scaldesian | represented by the marine | | | i | groups of Belgium and Holland; Sicilian | Floridian series; in the | | | o | and Astian of France and Italy. | interior by a subaerial and | | | c | Older:--English Coralline Crag; Diestian | lacustrine series; and on the | | | e | of Belgium; Plaisancian of southern | Pacific border by the thick | | | n | France and Italy. | marine series of San Francisco.| | | e | | | | | . | | | | +---+-------------------------------------------+----------------------------------+ | | M | Wanting in Britain; well developed in | Represented in the Eastern States| | | i | France, S. E. Europe and Italy; | by a marine series (Yorktown or| | C | o | divisible into the following groups in | Chesapeake, Chipola and | | a | c | descending order: (1) Pontian; (2) | Chattahoochee groups), and in | | i | e | Sarmatian; (3) Tortonian; (4) Helvetian;| the interior by the lacustrine | | n | n | (5) Langhian (Burdigalian). | Loup Fork (Nebraska), Deep | | o | e | | River, and John Day groups. | | z | . | | | | o +---+-------------------------------------------+----------------------------------+ | i | | In Britain the "fluvio-marine series" of | On the Atlantic border no | | c | O | the Isle of Wight; also the volcanic | equivalents have been | | | l | plateaux of Antrim and Inner Hebrides | satisfactorily recognised, but | | o | i | and those of the Faeroe Isles and | on the Pacific side there are | | r | g | Iceland. In continental Europe the | marine deposits in N. W. | | | o | following subdivisions have been | Oregon, which may represent | | T | c | established in descending order: (1) | this division. In the interior | | e | e | Aquitanian, (2) Stampian (Rupelian), | the equivalent is believed to | | r | n | (3) Tongrain (Sannoisian). | be the fresh-water White River | | t | e | | series, including (1) | | i | . | | _Protoceras_ beds, (2) | | a | | | _Oreodon_ beds, and (3) | | r | | | _Titanothervum_ beds. | | y +---+-------------------------------------------+----------------------------------+ | . | | Barton sands and clays; Ludian series of | Woodstock and Aquia Creek groups | | | | France. | of Potomac River; Vicksburg, | | | | Bracklesham Beds; Lutetian (Calcaire | Jackson, Claiborne, Buhrstone, | | | E | grossier and Caillasses) of Paris | and Lignitic groups of | | | o | basin. | Mississippi. | | | c | London clay, Woolwich and Reading Beds; | In the interior a thick series of| | | e | Thanet sands; Ypresian or Londinian of | fresh-water formations, | | | n | N. France and Belgium; Sparnacian and | comprising, in descending | | | e | Thanetian groups. | order, the Uinta, Bridger, | | | . | | Wind River, Wasatch, Torrejon, | | | | | and Puerco groups. | | | | | On the Pacific side the marine | | | | | Tejon series of Oregon and | | | | | California. | |---+---+-------------------------------------------+----------------------------------| | | | Upper | On the Atlantic border both | | | | ===== | marine strata and others | | | | Danian--wanting in Britain; uppermost | containing a terrestrial flora | | | | limestone of Denmark. | represent the Cretaceous series| | | | Senonian--Upper Chalk with Flints of | of formations. | | | | England; Aturian and Emscherian stages | In the interior there is also a | | | | on the European continent. | commingling of marine with | | | | Turonian--Middle Chalk with few flints, | lacustrine deposits. At the top| | | | and comprising the Angoumian and stages.| lies the Laramie or Lignitic | | | C | Cenomanian--Lower Chalk and Chalk Marl. | series with an abundant | | | r | | terrestrial flora, passing down| | | e | Lower | into the lacustrine and | | | t | ===== | brackish-water Montana series. | | | a | Albian--Upper Greensand and Gault. | Of older date, the Colorado | | | c | Aptian--Lower Greensand; Marls and | series contains an abundant | | | e | limestones of Provence, &c. | marine fauna, yet includes also| | | o | Urgonian (Barremian)--Atherfield clay; | some Niobrara marls and | | | u | massive Hippurite limestones of | limestones are likewise of | | | s | southern France. | marine origin, but the lower | | | . | Neocomian--Weald clay and Hastings sand; | members of the series (Benton | | | | Hauterivian and Valanginian sub-stages | and Dakota) show another great | | | | of Switzerland and France. | representation of fresh-water | | M | | | sedimentation with lignites and| | e | | | coals. | | s | | | In California a vast succession | | o | | | of marine deposits (Shasta- | | z | | | Chico) represents the | | o | | | Cretaceous system; and in | | i | | | western British N. America | | c | | | coal-seams also occur. | | +---+-------------------------------------------+----------------------------------+ | o | | Purbeckian--Purbeck beds; Munder Mergel; | Representatives of the Middle and| | r | | largely present in Westphalia. | lower Jurassic formations have | | | | Portlandian--Portland group of England, | been found in California and | | S | | represented in S. France by the thick | Oregon, and farther north among| | e | | Tithonian limestones. | the Arctic islands. | | c | | Kimmeridgian--Kimmeridge Clay of England; | Strata containing Lower Jurassic | | o | | Virgulian and Pterocerian groups of N. | marine fossils appear in | | n | J | France; represented by thick limestones | Wyoming and Dakota; and above | | d | u | in the Mediterranean basin. | them come the _Atlantosaurus_ | | a | r | Corallian--Coral Rag, Coralline Oolite; | and _Baptanodon_ beds, which | | r | a | Sequanian stages of the Continent, | have yielded so large a | | y | s | comprising the sub-stages of Astartian | variety of deinosaurs and other| | . | s | and Rauracian. | vertebrates, and especially the| | | i | Oxfordian--Oxford Clay; Axgovian and | remains of a number of genera | | | c | Neuvizyan stages. | of small mammals. | | | . | Callovian--Kellaways Rock, Divesian | | | | | sub-stage of N. France. | | | | | Bathonian--series of English strata from | | | | | Cornbrash down to Fuller's Earth. | | | | | Bajocian--Inferior Oolite of England. | | | | | Lassic--divisible into (1) Upper Lias | | | | | or Toarcian, (2) Middle Lias, Marlstone | | | | | or Charmouthian, (3) Lower Lias of | | | | | Sinemurian and Hettangian. | | | +---+-------------------------------------------+----------------------------------+ | | | In Germany and western Europe this | In New York, Connecticut, New | | | T | division represents the deposits of | Brunswick, and Nova Scotia | | | r | inland seas or lagoons, and is divisible| a series of red sandstone | | | i | into the following stages in descending | (Newark series) contains land- | | | a | order: (1) Rhaetic, (2) Keuper, (3) | plants and labyrinthodonts | | | s | Muschelkalk, (4) Bunter. In the | like the lagoon type of central| | | s | eastern Alps and the Mediterranean | and western Europe. On the | | | i | basin the contemporaneous sedimentary | Pacific slope, however, marine | | | c | formations are those of open clear | equivalents occur, representing| | | . | sea, in which a thickness of many | the pelagic type of south- | | | | thousand feet of strata was accumulated.| eastern Europe. | +---+---+-------------------------------------------+----------------------------------+ | | P | Thuringian--Zechstein, Magnesian | To this division of the geologi- | | | e | Limestone; named from its development | cal record the Upper Barren | | | r | in Thuringia; well represented | Measures of the coal-fields of | | | m | also in Saxony, Bavaria and Bohemia. | Pennsylvania, Prince Edward | | | i | Saxonian--Rothliegendes Group; Red | Island, Nova Scotia and | | | a | Sandstones, &c. | New Brunswick have been | | | n | Autunian--where the strata present the | assigned. | | | . | lagoon facies, well displayed at Autun | Farther south in Kansas, Texas, | | | | in France; where the marine type is | and Nebraska the representa- | | | | predominant, as in Russia, the group | tives of the division have an | | | | has been termed Artinskian. | abundant marine fauna. | | +---+-------------------------------------------+----------------------------------+ | | C | Stephanian or Uralian--represented in | Upper productive Coal-measures. | | | a | Russia by marine formations, and in | Lower Barren measures. | | | r | central and western Europe by numerous | Lower productive Coal-measures. | | | b | small basins containing a peculiar | Pottsville conglomerate. | | | o | flora and in some places a great variety| Mauch Chunk shales; limestones | | | n | of insects. | of Chester, St Louis, &c. | | | i | Westphalian or Moscovian--Coal-measures, | Pocono series; Kinderhook | | | f | Millstone Grit. | limestone. | | | e | Culm or Dinantian--Carboniferous Limestone| | | | r | and Calciferous Sandstone series. | | | | o | | | | | u | | | | | s | | | | | . | | | | +---+-------------------------------------------+----------------------------------+ | | Devonian and Old Red Sandstone. | | P +----------------------+------------------------+----------------------------------+ | a | Devonian type. | Old Red Sandstone | | | l | | type. | | | a +----------------------+------------------------+ / Catskill red sandstone; Old | | e | / Famennian. | Yellow and red | | Red Sandstone type: the | | o | Upper < | sandstone with |< strata below show the | | z | \ Frasnian. | _Holoptychius_, | | Devonian type. | | o | | _Bothriolepis_,&c. | | Chemung Group. | | i | | | \ Genesee " | | c | | | | | | / Givetian. | Caithness Flagstones | | | o | Middle < | with _Osteolepus_, | / Hamilton Group. | | r | \ Eifelian. | _Dipterus_, | \ Marcellus " | | | | _Homosteus_, &c. | | | P | | | | | r | | Red and purple | / Corniferous Limestone. / Upper | | i | /Coblentizian.| sandstones and | | | Held- | | m | Lower < | conglomerates with |< Onondaga Limestone. < erberg| | a | \Gedinnian. | _Cephalaspis_, | | \ Group.| | r | | _Pteraspis_, &c. | \ Oriskany Sandstone. | | y +---+------------------+------------------------+----------------------------------+ | . | | | / Lower Helderberg Group. | | | S | / Ludlow Group. | | Water-Lime. | | | i | Upper < Wenlock " | < Niagara Shale and Limestone. | | | l | \ Llandovery" | | Clinton Group. | | | u | | \ Medina " | | | r | | | | | i | | / Cincinnati Group. | | | a | Lower / Caradoc or Bala Group. | | Utica " | | | n | (Ordovician) < Llandeilo " | < Trenton " | | | . | \ Arenig " | | Chazy " | | | | | \ Calciferous " | | +---+-------------------------------------------+----------------------------------+ | | C | Upper or _Olenus_ series--Tremadoc | Upper or Potsdam series with | | | a | slates and _Lingula_ Flags. | _Olenus_ and _Dicelocephalus_ | | | m | Middle or _Pardoxides_ series--Menevian | fauna. | | | b | Group. | Middle or Acadian series with | | | r | Lower or _Olenellus_ series--Llanberis | _Paradoxides_ fauna. | | | i | and Harlech Group, and _Olenellus_- | Lower or Georgian series with | | | a | zone. | _Olenellus_ fauna. | | | n | | | | | . | | | +---+---+-------------------------------------------+----------------------------------+ | | | Archean, Pre-Cambrian, Eozoic. | +---+---+-------------------------------------------+----------------------------------+ | | | In Scotland, underneath the Cambrian | In Canada and the Lake Superior | | | | Olenellus group, lies unconformably | region of the United States | | | | a mass of red sandstone and con- | a vast succession of rocks of | | | | glomerate (Torridonian) 8000 or 10,000 | Pre-Cambrian age has been | | | | ft. thick, which rests with a strong | grouped into the following | | | | gneisses and schists (Lewisian). A | subdivisions in descending | | | | thick series of slates and phyllites | order: (1) Keweenwan, lying | | | | lies below the oldest Palaeozoic rocks | unconformably on (2) Animikie, | | | | in central Europe, with coarse | separated by a strong | | | | gneisses below. | unconformability from (3) Upper| | | | | Huronian, (4) Lower Huronian | | | | | with an unconformable base, (5)| | | | | Goutchiching, (6) Laurentian. | | | | | In the eastern part of Canada, | | | | | Newfoundland, &c., and also in | | | | | Montana, sedimentary formations| | | | | of great thickness below the | | | | | lowest Cambrian zone have been | | | | | found to contain some obscure | | | | | organisms. | +---+---+-------------------------------------------+----------------------------------+

From these general considerations we proceed to inquire how the existing topographical features of the land arose. Obviously the co-operation of the two great geological agencies of hypogene and epigene energy, which have been at work from the beginning of our globe's decipherable history, must have been the cause to which these features are to be assigned; and the task of the geologist is to ascertain, if possible, the part that has been taken by each. There is a natural tendency to see in a stupendous piece of scenery, such as a deep ravine, a range of hills, a line of precipice or a chain of mountains, evidence only of subterranean convulsion; and before the subject was taken up as a matter of strict scientific induction, an appeal to former cataclysms was considered a sufficient solution of the problems presented by such features of landscape. The rise of the modern Huttonian school, however, led to a more careful examination of these problems. The important share taken by erosion in the determination of the present features of landscape was then recognized, while a fuller appreciation of the relative parts played by the hypogene and epigene causes has gradually been reached.

1. The study of the progress of denudation at the present time has led to the conclusion that even if the rate of waste were not more rapid than it is to-day, it would yet suffice in a comparatively brief geological period to reduce the dry land to below the sea-level. But not only would the area of the land be diminished by denudation, it could hardly fail to be more or less involved in those widespread movements of subsidence, during which the thick sedimentary formations of the crust appear to have been accumulated. It is thus manifest that there must have been from time to time during the history of our globe upward movements of the crust, whereby the balance between land and sea was redressed. Proofs of such movements have been abundantly preserved among the stratified formations. We there learn that the uplifts have usually followed each other at long intervals between which subsidence prevailed, and thus that there has been a prolonged oscillation of the crust over the great continental areas of the earth's surface.

An examination of that surface leads to the recognition of two great types of upheaval. In the one, the sea-floor, with all its thick accumulations of sediment, has been carried upwards, sometimes for several thousand feet, so equably that the strata retain their original flatness with hardly any sensible disturbance for hundreds of square miles. In the other type the solid crust has been plicated, corrugated and dislocated, especially along particular lines, and has attained its most stupendous disruption in lofty chains of mountains. Between these two phases of uplift many intermediate stages have been developed, according to the direction and intensity of the subterranean force and the varying nature and disposition of the rocks Of the crust.

(a) Where the uplift has extended over wide spaces, without appreciable deformation of the crust, the flat strata have given rise to low plains, or if the amount of uprise has been great enough, to high plains, plateaux or tablelands. The plains of Russia, for example, lie for the most part on such tracts of equably uplifted strata. The great plains of the western interior of the United States form a great plateau or tableland, 5000 or 6000 ft. above the sea, and many thousands of square miles in extent, on which the Rocky Mountains have been ridged up.

(b) It is in a great mountain-chain that the complicated structures developed during disturbances of the earth's crust can best be studied (see Parts IV. and V. of this article), and where the influence of these structures on the topography of the surface is most effectively displayed. Such a chain may be the result of one colossal disturbance; but those of high geological antiquity usually furnish proofs of successive uplifts with more or less intervening denudation. Formed along lines of continental displacement in the crust, they have again and again given relief from the strain of compression by fresh crumpling, fracture and uprise. The chief guide in tracing these successive stages of growth is supplied by unconformability. If, for example, a mountain-range consists of upraised Silurian rocks, upon the upturned and denuded edges of which the Carboniferous Limestone lies transgressively, it is clear that its original upheaval must have taken place in the period of geological time represented by the interval between the Silurian and the Carboniferous Limestone formations. If, as the range is followed along its course, the Carboniferous Limestone is found to be also highly inclined and covered unconformably by the Upper Coal-measures, a second uplift of that portion of the ground can be proved to have taken place between the time of the Limestone and that of the Upper Coal-measures. By this simple and obvious kind of evidence the relative ages of different mountain-chains may be compared. In most great chains, however, the rocks have been so intensely crumpled, and even inverted, that much labour may be required before their true relations can be determined.

The Alps furnish an instructive example of the long series of revolutions through which a great mountain-system may have passed before reaching its present development. The first beginnings of the chain may have been upraised before the oldest Palaeozoic formations were laid down. There are at least traces of land and shore-lines in the Carboniferous period. Subsequent submergences and uplifts appear to have occurred during the Mesozoic periods. There is evidence that thereafter the whole region sank deep under the sea, in which the older Tertiary sediments were accumulated, and which seems to have spread right across the heart of the Old World. But after the deposition of the Eocene formations came the gigantic disruptions whereby all the rocks of the Alpine region were folded over each other, crushed, corrugated, fractured and displaced, some of their older portions, including the fundamental gneisses and schists, being squeezed up, torn off, and pushed horizontally for many miles over the younger rocks. But this upheaval, though the most momentous, was not the last which the chain has undergone, for at a later epoch in Tertiary time renewed disturbance gave rise to a further series of ruptures and plications. The chain thus successively upheaved has been continuously exposed to denudation and has consequently lost much of its original height. That it has been left in a state of instability is indicated by the frequent earthquakes of the Alpine region, which doubtless arise from the sudden snapping of rocks under intense strain.

A distinct type of mountain due to direct hypogene action is to be seen in a volcano. It has been already pointed out (Part IV. sect. 1) that at the vents which maintain a communication between the molten magma of the earth's interior and the surface, eruptions take place whereby quantities of lava and fragmentary materials are heaped round each orifice of discharge. A typical volcanic mountain takes the form of a perfect cone, but as it grows in size and its main vent is choked, while the sides of the cone are unable to withstand the force of the explosions or the pressure of the ascending column of lava, eruptions take place laterally, and numerous parasitic cones arise on the flanks of the parent mountain. Where lava flows out from long fissures, it may pile up vast sheets of rock, and bury the surrounding country under several thousand feet of solid stone, covering many hundreds of square miles. In this way volcanic tablelands have been formed which, attacked by the denuding forces, are gradually trenched by valleys and ravines, until the original level surface of the lava-field may be almost or wholly lost. As striking examples of this physiographical type reference may be made to the plateau of Abyssinia, the Ghats of India, the plateaux of Antrim, the Inner Hebrides and Iceland, and the great lava-plains of the western territories of the United States.

2. But while the subterranean movements have upraised portions of the surface of the lithosphere above the level of the ocean, and have thus been instrumental in producing the existing tracts of land, the detailed topographical features of a landscape are not solely, nor in general even chiefly, attributable to these movements. From the time that any portion of the sea-floor appears above sea-level, it undergoes erosion by the various epigene agents. Each climate and geological region has its own development of these agents, which include air, aridity, rapid and frequent alternations of wetness and dryness or of heat and cold, rain, springs, frosts, rivers, glaciers, the sea, plant and animal life. In a dry climate subject to great extremes of temperature the character and rate of decay will differ from those of a moist or an arctic climate. But it must be remembered that, however much they may vary in activity and in the results which they effect, the epigene forces work without intermission, while the hypogene forces bring about the upheaval of land only after long intervals. Hence, trifling as the results during a human life may appear, if we realize the multiplying influence of time we are led to perceive that the apparently feeble superficial agents can, in the course of ages, achieve stupendous transformations in the aspect of the land. If this efficacy may be deduced from what can be seen to be in progress now, it may not less convincingly be shown, from the nature of the sedimentary rocks of the earth's crust, to have been in progress from the early beginnings of geological history. Side by side with the various upheavals and subsidences, there has been a continuous removal of materials from the land, and an equally persistent deposit of these materials under water, with the consequent growth of new rocks. Denudation has been aptly compared to a process of sculpturing wherein, while each of the implements employed by nature, like a special kind of graving tool, produces its own characteristic impress on the land, they all combine harmoniously towards the achievement of their one common task. Hence the present contours of the land depend partly on the original configuration of the ground, and the influence it may have had in guiding the operations of the erosive agents, partly on the vigour with which these agents perform their work, and partly on the varying structure and powers of resistance possessed by the rocks on which the erosion is carried on.

Where a new tract of land has been raised out of the sea by such an energetic movement as broke up the crust and produced the complicated structure and tumultuous external forms of a great mountain chain, the influence of the hypogene forces on the topography attains its highest development. But even the youngest existing chain has suffered so greatly from denudation that the aspect which it presented at the time of its uplift can only be dimly perceived. No more striking illustration of this feature can be found than that supplied by the Alps, nor one where the geotectonic structures have been so fully studied in detail. On the outer flanks of these mountains the longitudinal ridges and valleys of the Jura correspond with lines of anticline and syncline. Yet though the dominant topographical elements of the region have obviously been produced by the plication of the stratified formations, each ridge has suffered so large an amount of erosion that the younger rocks have been removed from its crest where the older members of the series are now exposed to view, while on every slope proofs may be seen of extensive denudation. If from these long wave-like undulations of the ground, where the relations between the disposition of the rocks below and the forms of the surface are so clearly traceable, the observer proceeds inwards to the main chain, he finds that the plications and displacements of the various formations assume an increasingly complicated character; and that although proofs of great denudation continue to abound, it becomes increasingly difficult to form any satisfactory conjecture as to the shape of the ground when the upheaval ended or any reliable estimate of the amount of material which has since then been removed. Along the central heights the mountains lift themselves towards the sky like the storm-swept crests of vast earth-billows. The whole aspect of the ground suggests intense commotion, and the impression thus given is often much intensified by the twisted and crumpled strata, visible from a long distance, on the crags and crests. On this broken-up surface the various agents of denudation have been ceaselessly engaged since it emerged from the sea. They have excavated valleys, sometimes along depressions provided for them by the subterranean disturbances, sometimes down the slopes of the disrupted blocks of ground. So powerful has been this erosion that valleys cut out along lines of anticline, which were natural ridges, have sometimes become more important than those in lines of syncline, which were structurally depressions. The same subaerial forces have eroded lake-basins, dug out corries or cirques, notched the ridges, splintered the crests and furrowed the slopes, leaving no part of the original surface of the uplifted chain unmodified.

It has often been noted with surprise that features of underground structure which, it might have been confidently anticipated, should have exercised a marked influence on the topography of the surface have not been able to resist the levelling action of the denuding agents, and do not now affect the surface at all. This result is conspicuously seen in coal-fields where the strata are abundantly traversed by faults. These dislocations, having sometimes a displacement of several hundred feet, might have been expected to break up the surface into a network of cliffs and plains; yet in general they do not modify the level character of the ground above. One of the most remarkable faults in Europe is the great thrust which bounds the southern edge of the Belgian coal-field and brings the Devonian rocks above the Coal-measures. It can be traced across Belgium into the Boulonnais, and may not improbably run beneath the Secondary and Tertiary rocks of the south of England. It is crossed by the valleys of the Meuse and other northerly-flowing streams. Yet so indistinctly is it marked in the Meuse valley that no one would suspect its existence from any peculiarity in the general form of the ground, and even an experienced geologist, until he had learned the structure of the district, would scarcely detect any fault at all.

Where faults have influenced the superficial topography, it is usually by giving rise to a hollow along which the subaerial agents and especially running water can act effectively. Such a hollow may be eventually widened and deepened into a valley. On bare crags and crests, lines of fault are apt to be marked by notches or clefts, and they thus help to produce the pinnacles and serrated outlines of these exposed uplands.

It was cogently enforced by Hutton and Playfair, and independently by Lamarck, that no co-operation of underground agency is needed to produce such topography as may be seen in a great part of the world, but that if a tract of sea-floor were upraised into a wide plain, the fall of rain and the circulation of water over its surface would in the end carve out such a system of hills and valleys as may be seen on the dry land now. No such plain would be a dead-level. It would have inequalities on its surface which would serve as channels to guide the drainage from the first showers of rain. And these channels would be slowly widened and deepened until they would become ravines and valleys, while the ground between them would be left projecting as ridges and hills. Nor would the erosion of such a system of water-courses require a long series of geological periods for its accomplishment. From measurements and estimates of the amount of erosion now taking place in the basin of the Mississippi river it has been computed that valleys 800 ft. deep might be carved out in less than a million years. In the vast tablelands of Colorado and other western regions of the United States an impressive picture is presented of the results of mere subaerial erosion on undisturbed and nearly level strata. Systems of stream-courses and valleys, river gorges unexampled elsewhere in the world for depth and length, vast winding lines of escarpment, like ranges of sea-cliffs, terraced slopes rising from plateau to plateau, huge buttresses and solitary stacks standing like islands out of the plains, great mountain-masses towering into picturesque peaks and pinnacles cleft by innumerable gullies, yet everywhere marked by the parallel bars of the horizontal strata out of which they have been carved--these are the orderly symmetrical characteristics of a country where the scenery is due entirely to the action of subaerial agents on the one hand and the varying resistance of perfectly regular stratified rocks on the other.

The details of the sculpture of the land have mainly depended on the nature of the materials on which nature's erosive tools have been employed. The joints by which all rocks are traversed have been especially serviceable as dominant lines down which the rain has filtered, up which the springs have risen and into which the frost wedges have been driven. On the high bare scarps of a lofty mountain the inner structure of the mass is laid open, and there the system of joints even more than faults is seen to have determined the lines of crest, the vertical walls of cliff and precipice, the forms of buttress and recess, the position of cleft and chasm, the outline of spire and pinnacle. On the lower slopes, even under the tapestry of verdure which nature delights to hang where she can over her naked rocks, we may detect the same pervading influence of the joints upon the forms assumed by ravines and crags. Each kind of stone, too, gives rise to its own characteristic form of scenery. Massive crystalline rocks, such as granite, break up along their joints and often decay into sand or earth along their exposed surfaces, giving rise to rugged crags with long talus slopes at their base. The stratified rocks besides splitting at their joints are especially distinguished by parallel ledges, cornices and recesses, produced by the irregular decay of their component strata, so that they often assume curiously architectural types of scenery. But besides this family feature they display many minor varieties of aspect according to their lithological composition. A range of sandstone hills, for example, presents a marked contrast to one of limestone, and a line of chalk downs to the escarpments formed by alternating bands of harder and softer clays and shales.

It may suffice here merely to allude to a few of the more important parts of the topography of the land in their relation to physiographical geology. A true mountain-chain, viewed from the geological side, is a mass of high ground which owes its prominence to a ridging-up of the earth's crust, and the intense plication and rupture of the rocks of which it is composed. But ranges of hills almost mountainous in their bulk may be formed by the gradual erosion of valleys out of a mass of original high ground, such as a high plateau or tableland. Eminences which have been isolated by denudation from the main mass of the formations of which they originally formed part are known as "outliers" or "hills of circumdenudation."

Tablelands, as already pointed out, may be produced either by the upheaval of tracts of horizontal strata from the sea-floor into land; or by the uprise of plains of denudation, where rocks of various composition, structure and age have been levelled down to near or below the level of the sea by the co-operation of the various erosive agents. Most of the great tablelands of the globe are platforms of little-disturbed strata which have been upraised bodily to a considerable elevation. No sooner, however, are they placed in that position than they are attacked by running water, and begin to be hollowed out into systems of valleys. As the valleys sink, the platforms between them grow into narrower and more definite ridges, until eventually the level tableland is converted into a complicated network of hills and valleys, wherein, nevertheless, the key to the whole arrangement is furnished by a knowledge of the disposition and effects of the flow of water. The examples of this process brought to light in Colorado, Wyoming, Nevada and the other western regions by Newberry, King, Hayden, Powell and other explorers, are among the most striking monuments of geological operations in the world.

Examples of ancient and much decayed tablelands formed by the denudation of much disturbed rocks are furnished by the Highlands of Scotland and of Norway. Each of these tracts of high ground consists of some of the oldest and most dislocated formations of Europe, which at a remote period were worn down into a plain, and in that condition may have lain long submerged under the sea and may possibly have been overspread there with younger formations. Having at a much later time been raised several thousand feet above sea-level the ancient platforms of Britain and Scandinavia have been since exposed to denudation, whereby each of them has been so deeply channeled into glens and fjords that it presents to-day a surface of rugged hills, either isolated or connected along the flanks, while only fragments of the general surface of the tableland can here and there be recognized amidst the general destruction.

Valleys have in general been hollowed out by the greater erosive action of running water along the channels of drainage. Their direction has been probably determined in the great majority of cases by irregularities of the surface along which the drainage flowed on the first emergence of the land. Sometimes these irregularities have been produced by folds of the terrestrial crust, sometimes by faults, sometimes by the irregularities on the surface of an uplifted platform of deposition or of denudation. Two dominant trends may be observed among them. Some are longitudinal and run along the line of flexures in the upraised tract of land, others are transverse where the drainage has flowed down the slopes of the ridges into the longitudinal valleys or into the sea. The forms of valleys have been governed partly by the structure and composition of the rocks, and partly by the relative potency of the different denuding agents. Where the influence of rain and frost has been slight, and the streams, supplied from distant sources, have had sufficient declivity, deep, narrow, precipitous ravines or gorges have been excavated. The canyons of the arid region of the Colorado are a magnificent example of this result. Where, on the other hand, ordinary atmospheric action has been more rapid, the sides of the river channels have been attacked, and open sloping glens and valleys have been hollowed out. A gorge or defile is usually due to the action of a waterfall, which, beginning with some abrupt declivity or precipice in the course of the river when it first commenced to flow, or caused by some hard rock crossing the channel, has eaten its way backward.

Lakes have been already referred to, and their modes of origin have been mentioned. As they are continually being filled up with the detritus washed into them from the surrounding regions they cannot be of any great geological antiquity, unless where by some unknown process their basins are from time to time widened and deepened.

In the general subaerial denudation of a country, innumerable minor features are worked out as the structure of the rocks controls the operations of the eroding agents. Thus, among comparatively undisturbed strata, a hard bed resting upon others of a softer kind is apt to form along its outcrop a line of cliff or escarpment. Though a long range of such cliffs resembles a coast that has been worn by the sea, it may be entirely due to mere atmospheric waste. Again, the more resisting portions of a rock may be seen projecting as crags or knolls. An igneous mass will stand out as a bold hill from amidst the more decomposable strata through which it has risen. These features, often so marked on the lower grounds, attain their most conspicuous development among the higher and barer parts of the mountains, where subaerial disintegration is most rapid. The torrents tear out deep gullies from the sides of the declivities. Corries or cirques are scooped out on the one hand and naked precipices are left on the other. The harder bands of rock project as massive ribs down the slopes, shoot up into prominent _aiguilles_, or help to give to the summits the notched saw-like outlines they so often present.

The materials worn from the surface of the higher are spread out over the lower grounds. The streams as they descend begin to drop their freight of sediment when, by the lessening of their declivity, their carrying power is diminished. The great plains of the earth's surface are due to this deposit of gravel, sand and loam. They are thus monuments at once of the destructive and reproductive processes which have been in progress unceasingly since the first land rose above the sea and the first shower of rain fell. Every pebble and particle of their soil, once part of the distant mountains, has travelled slowly and fitfully to lower levels. Again and again have these materials been shifted, ever moving downward and sea-ward. For centuries, perhaps, they have taken their share in the fertility of the plains and have ministered to the nurture of flower and tree, of the bird of the air, the beast of the field and of man himself. But their destiny is still the great ocean. In that bourne alone can they find undisturbed repose, and there, slowly accumulating in massive beds, they will remain until, in the course of ages, renewed upheaval shall raise them into future land, there once more to pass through the same cycle of change. (A. Ge.)

LITERATURE.--_Historical_: The standard work is Karl A. von Zittel's _Geschichte der Geologie und Palaontologie_ (1899), of which there is an abbreviated, but still valuable, English translation; D'Archiac, _Histoire des progres de la geologie_, deals especially with the period 1834-1850; Keferstein, _Geschichte und Literatur der Geognosie_, gives a summary up to 1840; while Sir A. Geikie's _Founders of Geology_ (1897; 2nd ed., 1906) deals more particularly with the period 1750-1820. General treatises: Sir Charles Lyell's _Principles of Geology_ is a classic. Of modern English works, Sir A. Geikie's _Text Book of Geology_ (4th ed., 1903) occupies the first place; the work of T.C. Chamberlin and R.D. Salisbury, _Geology; Earth History_ (3 vols., 1905-1906), is especially valuable for American geology. A. de Lapparent's _Traite de geologie_ (5th ed., 1906), is the standard French work. H. Credner's _Elemente der Geologie_ has gone through several editions in Germany. Dynamical and physiographical geology are elaborately treated by E. Suess, _Das Antlitz der Erde_, translated into English, with the title _The Face of the Earth_. The practical study of the science is treated of by F. von Richthofen, _Fuhrer fur Forschungsreisende_ (1886); G.A. Cole, _Aids in Practical Geology_ (5th ed., 1906); A. Geikie, _Outlines of Field Geology_ (5th ed., 1900). The practical applications of Geology are discussed by J.V. Elsden, _Applied Geology_ (1898-1899). The relations of Geology to scenery are dealt with by Sir A. Geikie, _Scenery of Scotland_ (3rd ed., 1901); J.E. Marr, _The Scientific Study of Scenery_ (1900); Lord Avebury, _The Scenery of Switzerland_ (1896); _The Scenery of England_ (1902); and J. Geikie, _Earth Sculpture_ (1898). A detailed bibliography is given in Sir A. Geikie's _Text Book of Geology_. See also the separate articles on geological subjects for special references to authorities.

FOOTNOTES:

[1] In De Luc's _Lettres physiques et morales sur les montagnes_ (1778), the word "cosmology" is used for our science, the author stating that "geology" is more appropriate, but it "was not a word in use." In a completed edition, published in 1779, the same statement is made, but "geology" occurs in the text; in the same year De Saussure used the word without any explanation, as if it were well known.

[2] The subject of the age of the earth has also been discussed by Professor J. Joly and Professor W.J. Sollas. The former geologist, approaching the question from a novel point of view, has estimated the total quantity of sodium in the water of the ocean and the quantity of that element received annually by the ocean from the denudation of the land. Dividing the one sum by the other, he arrives at the result that the probable age of the earth is between 90 and 100 millions of years (_Trans. Roy. Dublin Soc._ ser. ii. vol. vii., 1899, p. 23: _Geol. Mag._, 1900, p. 220). Professor Sollas believes that this limit exceeds what is required for the evolution of geological history, that the lower limit assigned by Lord Kelvin falls short of what the facts demand, and that geological time will probably be found to have been comprised within some indeterminate period between these limits. (Address to Section C, _Brit. Assoc. Report_, 1900; _Age of the Earth_, London, 1905.)

GEOMETRICAL CONTINUITY. In a report of the Institute prefixed to Jean Victor Poncelet's _Traite des proprietes projectives des figures_ (Paris, 1822), it is said that he employed "ce qu'il appelle le principe de continuite." The law or principle thus named by him had, he tells us, been tacitly assumed as axiomatic by "les plus savans geometres." It had in fact been enunciated as "lex continuationis," and "la loi de la continuite," by Gottfried Wilhelm Leibnitz (Oxf. N.E.D.), and previously under another name by Johann Kepler in cap. iv. 4 of his _Ad Vitellionem paralipomena quibus astronomiae pars optica traditur_ (Francofurti, 1604). Of sections of the cone, he says, there are five species from the "recta linea" or line-pair to the circle. From the line-pair we pass through an infinity of hyperbolas to the parabola, and thence through an infinity of ellipses to the circle. Related to the sections are certain remarkable points which have no name. Kepler calls them foci. The circle has one focus at the centre, an ellipse or hyperbola two foci equidistant from the centre. The parabola has one focus within it, and another, the "caecus focus," which may be imagined to be _at infinity_ on the axis _within or without the curve_. The line from it to any point of the section is parallel to the axis. To carry out the analogy we must speak paradoxically, and say that the line-pair likewise has foci, which in this case coalesce as in the circle and fall upon the lines themselves; for our geometrical terms should be subject to analogy. Kepler dearly loves analogies, his most trusty teachers, acquainted with all the secrets of nature, "_omnium naturae arcanorum conscios_." And they are to be especially regarded in geometry as, by the use of "however absurd expressions," classing extreme limiting forms with an infinity of intermediate cases, and placing the whole essence of a thing clearly before the eyes.

Here, then, we find formulated by Kepler the doctrine of the concurrence of parallels at a single point at infinity and the principle of continuity (under the name analogy) in relation to the infinitely great. Such conceptions so strikingly propounded in a famous work could not escape the notice of contemporary mathematicians. Henry Briggs, in a letter to Kepler from Merton College, Oxford, dated "10 Cal. Martiis 1625," suggests improvements in the _Ad Vitellionem paralipomena_, and gives the following construction: Draw a line CBADC, and let an ellipse, a parabola, and a hyperbola have B and A for focus and vertex. Let CC be the other foci of the ellipse and the hyperbola. Make AD equal to AB, and with centres CC and radius in each case equal to CD describe circles. Then any point of the ellipse is equidistant from the focus B and one circle, and any point of the hyperbola from the focus B and the other circle. Any point P of the parabola, in which the second focus is missing or infinitely distant, is equidistant from the focus B and the line through D which we call the directrix, this taking the place of either circle when its centre C is at infinity, and every line CP being then parallel to the axis. Thus Briggs, and we know not how many "savans geometres" who have left no record, had already taken up the new doctrine in geometry in its author's lifetime. Six years after Kepler's death in 1630 Girard Desargues, "the Monge of his age," brought out the first of his remarkable works founded on the same principles, a short tract entitled _Methode universelle de mettre en perspective les objets donnes reellement ou en devis_ (Paris, 1636); but "Le privilege etoit de 1630." (Poudra, _[OE]uvres de Des._, i. 55). Kepler as a modern geometer is best known by his _New Stereometry of Wine Casks_ (Lincii, 1615), in which he replaces the circuitous Archimedean method of exhaustion by a direct "royal road" of infinitesimals, treating a vanishing arc as a straight line and regarding a curve as made up of a succession of short chords. Some 2000 years previously one Antipho, probably the well-known opponent of Socrates, has regarded a circle in like manner as the limiting form of a many-sided inscribed rectilinear figure. Antipho's notion was rejected by the men of his day as unsound, and when reproduced by Kepler it was again stoutly opposed as incapable of any sort of geometrical demonstration--not altogether without reason, for it rested on an assumed law of continuity rather than on palpable proof.

To complete the theory of continuity, the one thing needful was the idea of imaginary points implied in the algebraical geometry of Rene Descartes, in which equations between variables representing co-ordinates were found often to have imaginary roots. Newton, in his two sections on "Inventio orbium" (_Principia_ i. 4, 5), shows in his brief way that he is familiar with the principles of modern geometry. In two propositions he uses an auxiliary line which is supposed to cut the conic in X and Y, but, as he remarks at the end of the second (prop. 24), it may not cut it at all. For the sake of brevity he passes on at once with the observation that the required constructions are evident from the case in which the line cuts the trajectory. In the scholium appended to prop. 27, after saying that an asymptote is a tangent at infinity, he gives an unexplained general construction for the axes of a conic, which seems to imply that it has asymptotes. In all such cases, having equations to his loci in the background, he may have thought of elements of the figure as passing into the imaginary state in such manner as not to vitiate conclusions arrived at on the hypothesis of their reality.

Roger Joseph Boscovich, a careful student of Newton's works, has a full and thorough discussion of geometrical continuity in the third and last volume of his _Elementa universae matheseos_ (ed. prim. Venet, 1757), which contains _Sectionum conicarum elementa nova quadam methodo concinnata et dissertationem de transformatione locorum geometricorum, ubi de continuitatis lege, et de quibusdam infiniti mysteriis_. His first principle is that all varieties of a defined locus have the same properties, so that what is demonstrable of one should be demonstrable in like manner of all, although some artifice may be required to bring out the underlying analogy between them. The opposite extremities of an infinite straight line, he says, are to be regarded as joined, as if the line were a circle having its centre at the infinity on either side of it. This leads up to the idea of a _veluti plus quam infinita extensio_, a line-circle containing, as we say, the line infinity. Change from the real to the imaginary state is contingent upon the passage of some element of a figure through zero or infinity and never takes place _per saltum_. Lines being some positive and some negative, there must be negative rectangles and negative squares, such as those of the exterior diameters of a hyperbola. Boscovich's first principle was that of Kepler, by whose _quantumvis absurdis locutionibus_ the boldest applications of it are covered, as when we say with Poncelet that all concentric circles in a plane touch one another in two imaginary fixed points at infinity. In G.K. Ch. von Staudt's _Geometrie der Lage and Beitrage zur G. der L._ (Nurnberg, 1847, 1856-1860) the geometry of position, including the extension of the field of pure geometry to the infinite and the imaginary, is presented as an independent science, "welche des Messens nicht bedarf." (See GEOMETRY: _Projective_.)

Ocular illusions due to distance, such as Roger Bacon notices in the _Opus majus_ (i. 126, ii. 108, 497; Oxford, 1897), lead up to or illustrate the mathematical uses of the infinite and its reciprocal the infinitesimal. Specious objections can, of course, be made to the anomalies of the law of continuity, but they are inherent in the higher geometry, which has taught us so much of the "secrets of nature." Kepler's excursus on the "analogy" between the conic sections hereinbefore referred to is given at length in an article on "The Geometry of Kepler and Newton" in vol. xviii. of the _Transactions of the Cambridge Philosophical Society_ (1900). It had been generally overlooked, until attention was called to it by the present writer in a note read in 1880 (_Proc. C.P.S._ iv. 14-17), and shortly afterwards in _The Ancient and Modern Geometry of Conics, with Historical Notes and Prolegomena_ (Cambridge 1881). (C. T.*)

GEOMETRY, the general term for the branch of mathematics which has for its province the study of the properties of space. From experience, or possibly intuitively, we characterize existent space by certain fundamental qualities, termed axioms, which are insusceptible of proof; and these axioms, in conjunction with the mathematical entities of the point, straight line, curve, surface and solid, appropriately defined, are the premises from which the geometer draws conclusions. The geometrical axioms are merely conventions; on the one hand, the system may be based upon inductions from experience, in which case the deduced geometry may be regarded as a branch of physical science; or, on the other hand, the system may be formed by purely logical methods, in which case the geometry is a phase of pure mathematics. Obviously the geometry with which we are most familiar is that of existent space--the three-dimensional space of experience; this geometry may be termed Euclidean, after its most famous expositor. But other geometries exist, for it is possible to frame systems of axioms which definitely characterize some other kind of space, and from these axioms to deduce a series of non-contradictory propositions; such geometries are called non-Euclidean.

It is convenient to discuss the subject-matter of geometry under the following headings:

I. _Euclidean Geometry_: a discussion of the axioms of existent space and of the geometrical entities, followed by a synoptical account of Euclid's Elements.

II. _Projective Geometry_: primarily Euclidean, but differing from I. in employing the notion of geometrical continuity (q.v.)--points and lines at infinity.

III. _Descriptive Geometry_: the methods for representing upon planes figures placed in space of three dimensions.

IV. _Analytical Geometry_: the representation of geometrical figures and their relations by algebraic equations.

V. _Line Geometry_: an analytical treatment of the line regarded as the space element.

VI. _Non-Euclidean Geometry_: a discussion of geometries other than that of the space of experience.

VII. _Axioms of Geometry_: a critical analysis of the foundations of geometry.

Special subjects are treated under their own headings: e.g. PROJECTION, PERSPECTIVE; CURVE, SURFACE; CIRCLE, CONIC SECTION; TRIANGLE, POLYGON, POLYHEDRON; there are also articles on special curves and figures, e.g. ELLIPSE, PARABOLA, HYPERBOLA; TETRAHEDRON, CUBE, OCTAHEDRON, DODECAHEDRON, ICOSAHEDRON; CARDIOID, CATENARY, CISSOID, CONCHOID, CYCLOID, EPICYCLOID, LIMACON, OVAL, QUADRATRIX, SPIRAL, &c.

_History._--The origin of geometry (Gr. [Greek: ge], earth, [Greek: metron], a measure) is, according to Herodotus, to be found in the etymology of the word. Its birthplace was Egypt, and it arose from the need of surveying the lands inundated by the Nile floods. In its infancy it therefore consisted of a few rules, very rough and approximate, for computing the areas of triangles and quadrilaterals; and, with the Egyptians, it proceeded no further, the geometrical entities--the point, line, surface and solid--being only discussed in so far as they were involved in practical affairs. The point was realized as a mark or position, a straight line as a stretched string or the tracing of a pole, a surface as an area; but these units were not abstracted; and for the Egyptians geometry was only an art--an auxiliary to surveying.[1] The first step towards its elevation to the rank of a science was made by Thales (q.v.) of Miletus, who transplanted the elementary Egyptian mensuration to Greece. Thales clearly abstracted the notions of points and lines, founding the geometry of the latter unit, and discovering _per saltum_ many propositions concerning areas, the circle, &c. The empirical rules of the Egyptians were corrected and developed by the Ionic School which he founded, especially by Anaximander and Anaxagoras, and in the 6th century B.C. passed into the care of the Pythagoreans. From this time geometry exercised a powerful influence on Greek thought. Pythagoras (q.v.), seeking the key of the universe in arithmetic and geometry, investigated logically the principles underlying the known propositions; and this resulted in the formulation of definitions, axioms and postulates which, in addition to founding a _science_ of geometry, permitted a crystallization, fractional, it is true, of the amorphous collection of material at hand. Pythagorean geometry was essentially a geometry of areas and solids; its goal was the regular solids--the tetrahedron, cube, octahedron, dodecahedron and icosahedron--which symbolized the five elements of Greek cosmology. The geometry of the circle, previously studied in Egypt and much more seriously by Thales, was somewhat neglected, although this curve was regarded as the most perfect of all plane figures and the sphere the most perfect of all solids. The circle, however, was taken up by the Sophists, who made most of their discoveries in attempts to solve the classical problems of squaring the circle, doubling the cube and trisecting an angle. These problems, besides stimulating pure geometry, i.e. the geometry of constructions made by the ruler and compasses, exercised considerable influence in other directions. The first problem led to the discovery of the method of _exhaustion_ for determining areas. Antiphon inscribed a square in a circle, and on each side an isosceles triangle having its vertex on the circle; on the sides of the octagon so obtained, isosceles triangles were again constructed, the process leading to inscribed polygons of 8, 16 and 32 sides; and the areas of these polygons, which are easily determined, are successive approximations to the area of the circle. Bryson of Heraclea took an important step when he circumscribed, in addition to inscribing, polygons to a circle, but he committed an error in treating the circle as the mean of the two polygons. The method of Antiphon, in assuming that by continued division a polygon can be constructed coincident with the circle, demanded that magnitudes are not infinitely divisible. Much controversy ranged about this point; Aristotle supported the doctrine of infinite divisibility; Zeno attempted to show its absurdity. The mechanical tracing of loci, a principle initiated by Archytas of Tarentum to solve the last two problems, was a frequent subject for study, and several mechanical curves were thus discovered at subsequent dates (cissoid, conchoid, quadratrix). Mention may be made of Hippocrates, who, besides developing the known methods, made a study of similar figures, and, as a consequence, of proportion. This step is important as bringing into line discontinuous number and continuous magnitude.

A fresh stimulus was given by the succeeding Platonists, who, accepting in part the Pythagorean cosmology, made the study of geometry preliminary to that of philosophy. The many discoveries made by this school were facilitated in no small measure by the clarification of the axioms and definitions, the logical sequence of propositions which was adopted, and, more especially, by the formulation of the analytic method, i.e. of assuming the truth of a proposition and then reasoning to a known truth. The main strength of the Platonist geometers lies in stereometry or the geometry of solids. The Pythagoreans had dealt with the sphere and regular solids, but the pyramid, prism, cone and cylinder were but little known until the Platonists took them in hand. Eudoxus established their mensuration, proving the pyramid and cone to have one-third the content of a prism and cylinder on the same base and of the same height, and was probably the discoverer of a proof that the volumes of spheres are as the cubes of their radii. The discussion of sections of the cone and cylinder led to the discovery of the three curves named the parabola, ellipse and hyperbola (see CONIC SECTION); it is difficult to over-estimate the importance of this discovery; its investigation marks the crowning achievement of Greek geometry, and led in later years to the fundamental theorems and methods of modern geometry.

The presentation of the subject-matter of geometry as a connected and logical series of propositions, prefaced by [Greek: Horoi] or foundations, had been attempted by many; but it is to Euclid that we owe a complete exposition. Little indeed in the _Elements_ is probably original except the arrangement; but in this Euclid surpassed such predecessors as Hippocrates, Leon, pupil of Neocleides, and Theudius of Magnesia, devising an apt logical model, although when scrutinized in the light of modern mathematical conceptions the proofs are riddled with fallacies. According to the commentator Proclus, the _Elements_ were written with a twofold object, first, to introduce the novice to geometry, and secondly, to lead him to the regular solids; conic sections found no place therein. What Euclid did for the line and circle, Apollonius did for the conic sections, but there we have a discoverer as well as editor. These two works, which contain the greatest contributions to ancient geometry, are treated in detail in Section I. _Euclidean Geometry_ and the articles EUCLID; CONIC SECTION; APPOLONIUS. Between Euclid and Apollonius there flourished the illustrious Archimedes, whose geometrical discoveries are mainly concerned with the mensuration of the circle and conic sections, and of the sphere, cone and cylinder, and whose greatest contribution to geometrical method is the elevation of the method of exhaustion to the dignity of an instrument of research. Apollonius was followed by Nicomedes, the inventor of the conchoid; Diocles, the inventor of the cissoid; Zenodorus, the founder of the study of isoperimetrical figures; Hipparchus, the founder of trigonometry; and Heron the elder, who wrote after the manner of the Egyptians, and primarily directed attention to problems of practical surveying.

Of the many isolated discoveries made by the later Alexandrian mathematicians, those of Menelaus are of importance. He showed how to treat spherical triangles, establishing their properties and determining their congruence; his theorem on the products of the segments in which the sides of a triangle are cut by a line was the foundation on which Carnot erected his theory of transversals. These propositions, and also those of Hipparchus, were utilized and developed by Ptolemy (q.v.), the expositor of trigonometry and discoverer of many isolated propositions. Mention may be made of the commentator Pappus, whose _Mathematical Collections_ is valuable for its wealth of historical matter; of Theon, an editor of Euclid's _Elements_ and commentator of Ptolemy's _Almagest_; of Proclus, a commentator of Euclid; and of Eutocius, a commentator of Apollonius and Archimedes.

The Romans, essentially practical and having no inclination to study science _qua_ science, only had a geometry which sufficed for surveying; and even here there were abundant inaccuracies, the empirical rules employed being akin to those of the Egyptians and Heron. The Hindus, likewise, gave more attention to computation, and their geometry was either of Greek origin or in the form presented in trigonometry, more particularly connected with arithmetic. It had no logical foundations; each proposition stood alone; and the results were empirical. The Arabs more closely followed the Greeks, a plan adopted as a sequel to the translation of the works of Euclid, Apollonius, Archimedes and many others into Arabic. Their chief contribution to geometry is exhibited in their solution of algebraic equations by intersecting conics, a step already taken by the Greeks in isolated cases, but only elevated into a _method_ by Omar al Hayyami, who flourished in the 11th century. During the middle ages little was added to Greek and Arabic geometry. Leonardo of Pisa wrote a _Practica geometriae_ (1220), wherein Euclidean methods are employed; but it was not until the 14th century that geometry, generally Euclid's _Elements_, became an essential item in university curricula. There was, however, no sign of original development, other branches of mathematics, mainly algebra and trigonometry, exercising a greater fascination until the 16th century, when the subject again came into favour.

The extraordinary mathematical talent which came into being in the 16th and 17th centuries reacted on geometry and gave rise to all those characters which distinguish modern from ancient geometry. The first innovation of moment was the formulation of the principle of geometrical continuity by Kepler. The notion of infinity which it involved permitted generalizations and systematizations hitherto unthought of (see GEOMETRICAL CONTINUITY); and the method of indefinite division applied to rectification, and quadrature and cubature problems avoided the cumbrous method of exhaustion and provided more accurate results. Further progress was made by Bonaventura Cavalieri, who, in his _Geometria indivisibilibus continuorum_ (1620), devised a method intermediate between that of exhaustion and the infinitesimal calculus of Leibnitz and Newton. The logical basis of his system was corrected by Roberval and Pascal; and their discoveries, taken in conjunction with those of Leibnitz, Newton, and many others in the fluxional calculus, culminated in the branch of our subject known as differential geometry (see INFINITESIMAL CALCULUS; CURVE; SURFACE).

A second important advance followed the recognition that conics could be regarded as projections of a circle, a conception which led at the hands of Desargues and Pascal to modern _projective geometry_ and _perspective_. A third, and perhaps the most important, advance attended the application of algebra to geometry by Descartes, who thereby founded _analytical geometry_. The new fields thus opened up were diligently explored, but the calculus exercised the greatest attraction and relatively little progress was made in geometry until the beginning of the 19th century, when a new era opened.

Gaspard Monge was the first important contributor, stimulating analytical and differential geometry and founding _descriptive geometry_ in a series of papers and especially in his lectures at the Ecole polytechnique. Projective geometry, founded by Desargues, Pascal, Monge and L.N.M. Carnot, was crystallized by J.V. Poncelet, the creator of the modern methods. In his _Traite des proprietes des figures_ (1822) the line and circular points at infinity, imaginaries, polar reciprocation, homology, cross-ratio and projection are systematically employed. In Germany, A.F. Mobius, J. Plucker and J. Steiner were making far-reaching contributions. Mobius, in his _Barycentrische Calcul_ (1827), introduced homogeneous co-ordinates, and also the powerful notion of geometrical transformation, including the special cases of collineation and duality; Plucker, in his _Analytisch-geometrische Entwickelungen_ (1828-1831), and his _System der analytischen Geometrie_ (1835), introduced the abridged notation, line and plane co-ordinates, and the conception of generalized space elements; while Steiner, besides enriching geometry in numerous directions, was the first to systematically generate figures by projective pencils. We may also notice M. Chasles, whose _Apercu historique_ (1837) is a classic. Synthetic geometry, characterized by its fruitfulness and beauty, attracted most attention, and it so happened that its originally weak logical foundations became replaced by a more substantial set of axioms. These were found in the anharmonic ratio, a device leading to the liberation of synthetic geometry from metrical relations, and in involution, which yielded rigorous definitions of imaginaries. These innovations were made by K.J.C. von Staudt. Analytical geometry was stimulated by the algebra of invariants, a subject much developed by A. Cayley, G. Salmon, S.H. Aronhold, L.O. Hesse, and more particularly by R.F.A. Clebsch.

The introduction of the line as a space element, initiated by H. Grassmann (1844) and Cayley (1859), yielded at the hands of Plucker a new geometry, termed _line geometry_, a subject developed more notably by F. Klein, Clebsch, C.T. Reye and F.O.R. Sturm (see Section V., _Line Geometry_).

_Non-euclidean geometries_, having primarily their origin in the discussion of Euclidean parallels, and treated by Wallis, Saccheri and Lambert, have been especially developed during the 19th century. Four lines of investigation may be distinguished:--the naive-synthetic, associated with Lobatschewski, Bolyai, Gauss; the metric differential, studied by Riemann, Helmholtz, Beltrami; the projective, developed by Cayley, Klein, Clifford; and the critical-synthetic, promoted chiefly by the Italian mathematicians Peano, Veronese, Burali-Forte, Levi Civitta, and the Germans Pasch and Hilbert. (C. E.*)

I. EUCLIDEAN GEOMETRY

This branch of the science of geometry is so named since its methods and arrangement are those laid down in Euclid's _Elements_.

S 1. _Axioms._--The object of geometry is to investigate the properties of space. The first step must consist in establishing those fundamental properties from which all others follow by processes of deductive reasoning. They are laid down in the Axioms, and these ought to form such a system that nothing need be added to them in order fully to characterize space, and that nothing may be omitted without making the system incomplete. They must, in fact, completely "define" space.

S 2. _Definitions._--The axioms of Euclidean Geometry are obtained from inspection of existent space and of solids in existent space,--hence from experience. The same source gives us the notions of the geometrical entities to which the axioms relate, viz. solids, surfaces, lines or curves, and points. A solid is directly given by experience; we have only to abstract all material from it in order to gain the notion of a geometrical solid. This has shape, size, position, and may be moved. Its boundary or boundaries are called surfaces. They separate one part of space from another, and are said to have no thickness. Their boundaries are curves or lines, and these have length only. Their boundaries, again, are points, which have no magnitude but only position. We thus come in three steps from solids to points which have no magnitude; in each step we lose one extension. Hence we say a solid has three dimensions, a surface two, a line one, and a point none. Space itself, of which a solid forms only a part, is also said to be of three dimensions. The same thing is intended to be expressed by saying that a solid has length, breadth and thickness, a surface length and breadth, a line length only, and a point no extension whatsoever.

Euclid gives the essence of these statements as definitions:--

Def. 1, I. _A point is that which has no parts, or which has no magnitude._

Def. 2, I. _A line is length without breadth._

Def. 5, I. _A superficies is that which has only length and breadth._

Def. 1, XI. _A solid is that which has length, breadth and thickness._

It is to be noted that the synthetic method is adopted by Euclid; the analytical derivation of the successive ideas of "surface," "line," and "point" from the experimental realization of a "solid" does not find a place in his system, although possessing more advantages.

If we allow motion in geometry, we may generate these entities by moving a point, a line, or a surface, thus:--

The path of a moving point is a line.

The path of a moving line is, in general, a surface.

The path of a moving surface is, in general, a solid.

And we may then assume that the lines, surfaces and solids, as defined before, can all be generated in this manner. From this generation of the entities it follows again that the boundaries--the first and last position of the moving element--of a line are points, and so on; and thus we come back to the considerations with which we started.

Euclid points this out in his definitions,--Def. 3, I., Def. 6, I., and Def. 2, XI. He does not, however, show the connexion which these definitions have with those mentioned before. When points and lines have been defined, a statement like Def. 3, I., "The extremities of a line are points," is a proposition which either has to be proved, and then it is a theorem, or which has to be taken for granted, in which case it is an axiom. And so with Def. 6, I., and Def. 2, XI.

S 3. Euclid's definitions mentioned above are attempts to describe, in a few words, notions which we have obtained by inspection of and abstraction from solids. A few more notions have to be added to these, principally those of the simplest line--the straight line, and of the simplest surface--the flat surface or plane. These notions we possess, but to define them accurately is difficult. Euclid's Definition 4, I., "A straight line is that which lies evenly between its extreme points," must be meaningless to any one who has not the notion of straightness in his mind. Neither does it state a property of the straight line which can be used in any further investigation. Such a property is given in Axiom 10, I. It is really this axiom, together with Postulates 2 and 3, which characterizes the straight line.

Whilst for the straight line the verbal definition and axiom are kept apart, Euclid mixes them up in the case of the plane. Here the Definition 7, I., includes an axiom. It defines a plane as a surface which has the property that every straight line which joins any two points in it lies altogether in the surface. But if we take a straight line and a point in such a surface, and draw all straight lines which join the latter to all points in the first line, the surface will be fully determined. This construction is therefore sufficient as a definition. That every other straight line which joins any two points in this surface lies altogether in it is a further property, and to assume it gives another axiom.

Thus a number of Euclid's axioms are hidden among his first definitions. A still greater confusion exists in the present editions of Euclid between the postulates and axioms so called, but this is due to later editors and not to Euclid himself. The latter had the last three axioms put together with the postulates [Greek: (aitemata)], so that these were meant to include all assumptions relating to space. The remaining assumptions, which relate to magnitudes in general, viz. the first eight "axioms" in modern editions, were called "common notions" [Greek: (koivai ennoiai)]. Of the latter a few may be said to be definitions. Thus the eighth might be taken as a definition of "equal," and the seventh of "halves." If we wish to collect the axioms used in Euclid's _Elements_, we have therefore to take the three postulates, the last three axioms as generally given, a few axioms hidden in the definitions, and an axiom used by Euclid in the proof of Prop. 4, I, and on a few other occasions, viz. that figures may be moved in space without change of shape or size.

S 4. _Postulates._--The assumptions actually made by Euclid may be stated as follows:--

(1) Straight lines exist which have the property that any one of them may be produced both ways without limit, that through any two points in space such a line may be drawn, and that any two of them coincide throughout their indefinite extensions as soon as two points in the one coincide with two points in the other. (This gives the contents of Def. 4, part of Def. 35, the first two Postulates, and Axiom 10.)

(2) Plane surfaces or planes exist having the property laid down in Def. 7, that every straight line joining any two points in such a surface lies altogether in it.

(3) Right angles, as defined in Def. 10, are possible, and all right angles are equal; that is to say, wherever in space we take a plane, and wherever in that plane we construct a right angle, all angles thus constructed will be equal, so that any one of them may be made to coincide with any other. (Axiom 11.)

(4) The 12th Axiom of Euclid. This we shall not state now, but only introduce it when we cannot proceed any further without it.

(5) Figures maybe freely moved in space without change of shape or size. This is assumed by Euclid, but not stated as an axiom.

(6) In any plane a circle may be described, having any point in that plane as centre, and its distance from any other point in that plane as radius. (Postulate 3.)

The definitions which have not been mentioned are all "nominal definitions," that is to say, they fix a name for a thing described. Many of them overdetermine a figure.

S 5. Euclid's _Elements_ (see EUCLID) are contained in thirteen books. Of these the first four and the sixth are devoted to "plane geometry," as the investigation of figures in a plane is generally called. The 5th book contains the theory of proportion which is used in Book VI. The 7th, 8th and 9th books are purely arithmetical, whilst the 10th contains a most ingenious treatment of geometrical irrational quantities. These four books will be excluded from our survey. The remaining three books relate to figures in space, or, as it is generally called, to "solid geometry." The 7th, 8th, 9th, 10th, 13th and part of the 11th and 12th books are now generally omitted from the school editions of the _Elements_. In the first four and in the 6th book it is to be understood that all figures are drawn in a plane.