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 andthe 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 prominentaiguilles, 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 andhave 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’sGeschichte der Geologie und Paläontologie(1899), of which there is an abbreviated, but still valuable, English translation; D’Archiac,Histoire des progrès de la géologie, deals especially with the period 1834-1850; Keferstein,Geschichte und Literatur der Geognosie, gives a summary up to 1840; while Sir A. Geikie’sFounders of Geology(1897; 2nd ed., 1906) deals more particularly with the period 1750-1820. General treatises: Sir Charles Lyell’sPrinciples of Geologyis a classic. Of modern English works, Sir A. Geikie’sText 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’sTraité de géologie(5th ed., 1906), is the standard French work. H. Credner’sElemente der Geologiehas 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 titleThe Face of the Earth. The practical study of the science is treated of by F. von Richthofen,Führer für 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’sText Book of Geology. See also the separate articles on geological subjects for special references to authorities.
Literature.—Historical: The standard work is Karl A. von Zittel’sGeschichte der Geologie und Paläontologie(1899), of which there is an abbreviated, but still valuable, English translation; D’Archiac,Histoire des progrès de la géologie, deals especially with the period 1834-1850; Keferstein,Geschichte und Literatur der Geognosie, gives a summary up to 1840; while Sir A. Geikie’sFounders of Geology(1897; 2nd ed., 1906) deals more particularly with the period 1750-1820. General treatises: Sir Charles Lyell’sPrinciples of Geologyis a classic. Of modern English works, Sir A. Geikie’sText 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’sTraité de géologie(5th ed., 1906), is the standard French work. H. Credner’sElemente der Geologiehas 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 titleThe Face of the Earth. The practical study of the science is treated of by F. von Richthofen,Führer für 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’sText Book of Geology. See also the separate articles on geological subjects for special references to authorities.
1In De Luc’sLettres 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.2The 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.)
1In De Luc’sLettres 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.
2The 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’sTraité des propriétés projectives des figures(Paris, 1822), it is said that he employed “ce qu’il appelle le principe de continuité.” The law or principle thus named by him had, he tells us, been tacitly assumed as axiomatic by “les plus savans géomètres.” It had in fact been enunciated as “lex continuationis,” and “la loi de la continuité,” by Gottfried Wilhelm Leibnitz (Oxf. N.E.D.), and previously under another name by Johann Kepler in cap. iv. 4 of hisAd 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 beat infinityon the axiswithin 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 theAd 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 géomètres” 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 entitledMéthode universelle de mettre en perspective les objets donnés réellement ou en devis(Paris, 1636); but “Le privilége étoit de 1630.” (Poudra,Œuvres de Des., i. 55). Kepler as a modern geometer is best known by hisNew 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 René Descartes, in which equations between variables representing co-ordinates were found often to have imaginary roots. Newton, in his two sections on “Inventio orbium” (Principiai. 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 hisElementa universae matheseos(ed. prim. Venet, 1757), which containsSectionum 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 aveluti 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 placeper 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 whosequantumvis absurdis locutionibusthe boldestapplications 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’sGeometrie der Lage and Beiträge zur G. der L.(Nürnberg, 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.” (SeeGeometry:Projective.)
Ocular illusions due to distance, such as Roger Bacon notices in theOpus 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 theTransactions 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 inThe 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,Limaçon,Oval,Quadratrix,Spiral, &c.
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,Limaçon,Oval,Quadratrix,Spiral, &c.
History.—The origin of geometry (Gr.γῆ, earth,μέτρον, 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.1The 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 discoveringper saltummany 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 centuryB.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 ascienceof 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 ofexhaustionfor 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 aknown 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 (seeConic 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 byὍροιor foundations, had been attempted by many; but it is to Euclid that we owe a complete exposition. Little indeed in theElementsis 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, theElementswere 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 Geometryand the articlesEuclid;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, whoseMathematical Collectionsis valuable for its wealth of historical matter; of Theon, an editor of Euclid’sElementsand commentator of Ptolemy’sAlmagest; 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 sciencequascience, 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 amethodby 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 aPractica geometriae(1220), wherein Euclidean methods are employed; but it was not until the 14th century that geometry, generally Euclid’sElements, 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 (seeGeometrical 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 hisGeometria 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 (seeInfinitesimal 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 modernprojective geometryandperspective. A third, and perhaps the most important, advance attended the application of algebra to geometry by Descartes, who thereby foundedanalytical 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 foundingdescriptive geometryin a series of papers and especially in his lectures at the École 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 hisTraité des propriétés 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. Möbius, J. Plücker and J. Steiner were making far-reaching contributions. Möbius, in hisBarycentrische Calcul(1827), introduced homogeneous co-ordinates, and also the powerful notion of geometrical transformation, including the special cases of collineation and duality; Plücker, in hisAnalytisch-geometrische Entwickelungen(1828-1831), and hisSystem 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, whoseAperçu 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 byH. Grassmann (1844) and Cayley (1859), yielded at the hands of Plücker a new geometry, termedline 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 naïve-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 Civittà, and the Germans Pasch and Hilbert.