A the varying resistance of perfectly regular stratified rocks on the of Britain and Scandinavia have been since exposed to denudaother. tion, whereby each of them has been so deeply channeled into The details of the sculpture of the land have mainly depended glens and fjords that it presents to-day a surface of rugged on the nature of the materials on which nature's erosive tools hills, either isolated or connected along the flanks, while only have been employed. The joints by which all rocks are traversed fragments of the general surface of the tableland can here and have been especially serviceable as dominant lines down which there be recognized amidst the general destruction. the rain has filtered, up which the springs have risen and into Valleys have in general been hollowed out by the greater which the frost wedges have been driven. On the high bare erosive action of running water along the channels of drainage. scarps of a lofty mountain the inner structure of the mass is laid Their direction has been probably determined in the great open, and there the system of joints even more than faults is majority of cases by irregularities of the surface along which seen to have determined the lines of crest, the vertical walls of the drainage flowed on the first emergence of the land. Somecliff and precipice, the forms of buttress and recess, the position times these irregularities have been produced by folds of the of cleft and chasm, the outline of spire and pinnacle. On the terrestrial crust, sometimes by faults, sometimes by the irregulower slopes, even under the tapestry of verdure which nature larities on the surface of an uplifted platform of deposition or of delights to hang where she can over her naked rocks, we may denudation. Two dominant trends may be observed among detect the same pervading influence of the joints upon the forms them. Some are longitudinal and run along the line of flexures assumed by ravines and crags. Each kind of stone, too, gives in the upraised tract of land, others are transverse where the rise to its own characteristic form of scenery. Massive crystalline drainage has flowed down the slopes of the ridges into the longirocks, such as granite, break up along their joints and often tudinal valleys or into the sea. The forms of valleys have been decay into sand or earth along their exposed surfaces, giving governed partly by the structure and composition of the rocks, rise to rugged crags with long talus slopes at their base. Thc and partly by the relative potency of the different denuding stratified rocks besides splitting at their joints are especially agents. Where the influence of rain and frost has been slight, distinguished by parallel ledges, cornices and recesses, produced and the streams, supplied from distant sources, have had by the irregular decay of their component strata, so that they sufficient declivity, deep, narrow, precipitous ravines or gorges often assume curiously architectural types of scenery. But have been excavated. The canyons of the arid region of the besides this family feature they display many minor varieties of Colorado are a magnificent example of this result. Where, on aspect according to their lithological composition. A range of the other hand, ordinary atmospheric action has been more sandstone hills, for example, presents a marked contrast to one rapid, the sides of the river channels have been attacked, and of limestone, and a line of chalk downs to the escarpments open sloping glens and valleys have been hollowed out. formed by alternating bands of harder and softer clays and gorge or defile is usually due to the action of a waterfall, which, shales. beginning with some abrupt declivity or precipice in the course It may suffice here merely to allude to a few of the more of the river when it first commenced to flow, or caused by some important parts of the topography of the land in their relation hard rock crossing the channel, has eaten its way backward. to physiographical geology. A true mountain-chain, viewed Lakes have been already referred to, and their modes of origin from the geological side, is a mass of high ground which owes its have been mentioned. As they are continually being filled up prominence to a ridging-up of the earth's crust, and the intense with the detritus washed into them from the surrounding plication and rupture of the rocks of which it is composed. But regions they cannot be of any great geological antiquity, unless ranges of hills almost mountainous in their bulk may be formed where by some unknown process their basins are from time to by the gradual erosion of valleys out of a mass of original high time widened and deepened. ground, such as a high plateau or tableland. Eminences which In the general subaerial denudation of a country, innumerable have been isolated by denudation from the main mass of the minor fcatures are worked out as the structure of the rocks formations of which they originally formed part are known as controls the operations of the eroding agents. Thus, among "outliers" or " hills of circumdenudation." comparatively undisturbed strata, a hard bed resting upon Tablelands, as already pointed out, may be produced either others of a softer kind is apt to form along its outcrop a line of by the upheaval of tracts of horizontal strata from the sea-floor cliff or escarpment. Though a long range of such cliffs resembles into land; or by the uprise of plains of denudation, where rocks a coast that has been worn by the sea, it may be entirely due to of various composition, structure and age have been levelled mere atmospheric waste. Again, the more resisting portions of down to near or below the level of the sea by the co-operation a rock may be seen projecting as crags or knolls. An igneous of the various erosive agents. Most of the great tablelands mass will stand out as a bold hill from amidst the more decomof the globe are platforms of little-disturbed strata which have posable strata through which it has risen. These features been upraised bodily to a considerable elevation. No sooner, often so marked on the lower grounds, attain their most conhowever, are they placed in that position than they are attacked spicuous development among the higher and barer parts of the by running water, and begin to be hollowed out into systems of mountains, where subaerial disintegration is most rapid. The valleys. As the valleys sink, the platforms between them grow torrents tear out deep gullies from the sides of the declivities. into narrower and more definite ridges, until eventually the Corries or cirques are scooped out on the one hand and naked level tableland is converted into a complicated network of hills precipices are left on the other. The harder bands of rock and valleys, wherein, nevertheless, the key to the whole arrange- project as massive ribs down the slopes, shoot up into prominent ment is furnished by a knowledge of the disposition and effects aiguilles, or help to give to the summits the notched saw-like of the flow of water. The examples of this process brought to outlines they so often présent. light in Colorado, Wyoming, Nevada and the other western The materials worn from the surface of the higher are spread regions by Newberry, King, Hayden, Powell and other explorers, out over the lower grounds. The streams as they descend begin are among the most striking monuments of geological operations to drop their freight of sediment when, by the lessening of their in the world. declivity, their carrying power is diminished. The great plains Examples of ancient and much decayed tablelands formed by of the earth's surface are due to this deposit of gravel, sand and the denudation of much disturbed rocks are furnished by the loam. They are thus monuments at once of the destructive and Highlands of Scotland and of Norway. Each of these tracts of reproductive processes which have been in progress unceasingly high ground consists of some of the oldest and most dislocated since the first land rose above the sea and the first shower of rain formations of Europe, which at a remote period were worn down fell. Every pebble and particle of their soil, once part of the into a plain, and in that condition may have lain long submerged distant mountains, has travelled slowly and fitfully to lower under the sca and may possibly have been overspread there levels. Again and again have these materials been shifted, with younger formations. Having at a much later time been ever moving downward and sea-ward. For centuries, perhaps, raised several thousand feet above sea level the ancient platforms they have taken their share in the fertility of the plains and how many have ministered to the nurture of flower and tree, of the bird of vertex. Let CC be the other foci of the ellipse and the hyperbola. the air, the beast of the field and of man himself. But their Make AD equal to AB, and with centres CC and radius in each destiny is still the great ocean. In that bourne alone can they case equal to CD describe circles. Then any point of the ellipse find undisturbed repose, and there, slowly accumulating in is equidistant from the focus B and one circle, and any point of massive beds, they will remain until, in the course of ages, the hyperbola from the focus B and the other circle. Any point renewed upheaval shall raise them into future land, there once P of the parabola, in which the second focus is missing or inmore to pass through the same cycle of change. (A. GE.) finitely distant, is equidistant from the focus B and the line - LITERATURE.--Historical: The standard work is Karl A. von through D which we call the directrix, this taking the place of Zittel's Geschichte der Geologie und Paläontologie (1899), of which either circle when its centre C is at infinity, and every line CP there is an abbreviated, but still valuable, English translation; being then parallel to the axis. Thus Briggs, and we know not D'Archiac, Histoire des progrès de la géologie, deals especially with the period 1834-1850; Keferstein, Geschichte Literatur der savans géomètres " who have left no record, had Geognosie, gives a summary up to 1840; while Sir A. Geikie's already taken up the new doctrine in geometry in its author's Founders of Geology (1897; 2nd ed., 1906) deals more particularly lifetime. Six years after Kepler's death in 1630 Girard Desargues, with the period 1750-1820. General treatises: Sir Charles Lyell's “the Monge of his age," brought out the first of his remarkable 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; works founded on the same principles, a short tract entitled the work of T. C. Chamberlin and R. P. Salisbury, Geology; Earth Méthode universelle de mettre en perspective les objets donnés History (3 vols., 1905-1906), is especially valuable for American réellement ou en devis (Paris, 1636); but “Le privilége étoit de geology. A. de Lapparent's Traile de géologie (5th ed., 1906), is the 1630" (Poudra, Euvres de Des., i. 55). Kepler as a modern standard French work. H. Credner's Elemente der Geologie has gone through several editions in Germany. Dynamical and physio- geometer is best known by his New Stereomeiry of Wine Casks graphical geology are elaborately treated by E. Suess, Das Antlitz (Lincii, 1615), in which he replaces the circuitous Archimedean der Erde, translated into English, with the title The Face of the Earth. method of exhaustion by a direct "royal road " of infinitesimals, The practical study of the science is treated of by F. von Richthofen, treating a vanishing arc as a straight line and regarding a curve Führer für Forschungsreisende (1886); G. A. Cole, Aids in Practical as made up of a succession of short chords. Some 2000 years Gealogy (5th ed., 1906); A. Geikie, Outlines of Field Geology (5th ed., 1900). The practical applications of Geology are discussed by previously one Antipho, probably the well-known opponent of j.V. Elsden, Applied Geology (1898–1899). The relations of Geology Socrates, has regarded a circle in like manner as the limiting to scenery are dealt with by Sir A. Geikie, Scenery of Scotland (3rd ed. form of a many-sided inscribed rectilinear figure. Antipho's 1901); J. E. Marr, The Scientific Study of Scenery (1900): Lord notion was rejected by the men of his day as unsound, and when (1902); and J. Geikie, Earth Sculpture (1898). A detailed biblio- reproduced by Kepler it was again stoutly opposed as incapable graphy is given in Sir A. Geikie's Text Book of Geology, See also of any sort of geometrical demonstration—not altogether withthe separate articles on geological subjects for special references to out reason, for it rested on an assumed law of continuity rather authorities. than on palpable proof. GEOMETRICAL CONTINUITY. In a report of the Institute To complete the theory of continuity, the one thing needful prefixed to Jean Victor Poncelet's Traité des propriétés pro- was the idea of imaginary points implied in the algebraical jectives des figures (Paris, 1822), it is said that he employed " ce geometry of René Descartes, in which equations between variqu'il appelle le principe de continuité.” The law or principle ables representing co-ordinates were found often to have imaginary thus named by him had, he tells us, been tacitly assumed as roots. Newton, in his two sections on " Inventio orbium" axiomatic by " les plus savans géomètres." It had in fact been (Principia i. 4, 5), shows in his brief way that he is familiar with enunciated as “ lex continuationis," and“ la loi de la continuité,” the principles of modern geometry. In two propositions he uses by Gottfried Wilhelm Leibnitz (Oxf. N.E.D.), and previously an auxiliary line which is supposed to cut the conic in X and Y, under another name by Johann Kepler in cap. iv. 4 of his Ad but, as he remarks at the end of the second (prop. 24), it may not Vilellionem paralipomena quibus astronomiae pars oplica traditur cut it at all. For the sake of brevity he passes on at once with the (Francofurti, 1604). Of sections of the cone, he says, there are observation that the required constructions are evident from the five species from the “recta linea " or line-pair to the circle. case in which the line cuts the trajectory. In the scholium From the line-pair we pass through an infinity of hyperbolas to appended to prop. 27, after saying that an asymptote is a tangent the parabola, and thence through an infinity of ellipses to the at infinity, he gives an unexplained general construction for the circle. Related to the sections are certain remarkable points axes of a conic, which seems to imply that it has asymptotes. which have no name. Kepler calls them foci. The circle has In all such cases, having equations to his loci in the background, one focus at the centre, an ellipse or hyperbola two foci equi- he may have thought of elements of the figure as passing into the distant from the centre. The parabola has one focus within it, imaginary state in such manner as not to vitiate conclusions and another, the “caecus focus," which may be imagined to be arrived at on the hypothesis of their reality. at infinity on the axis within or wilhout the curve. The line from it Roger Joseph Boscovich, a careful student of Newton's works, to any point of the section is parallel to the axis. To carry out has a full and thorough discussion of geometrical continuity in the analogy we must speak paradoxically, and say that the line the third and last volume of his Elementa universae matheseos pair likewise has foci, which in this case coalesce as in the circle (ed. prim. Venet, 1757), which contains Sectionum conicarum and fall upon the lines themselves; for our geometrical terms elementa nova quadam methodo concinnata el dissertationem de should be subject to analogy. Kepler dearly loves analogies, his transformatione locorum geometricorum, ubi de continuitatis most trusty teachers, acquainted with all the secrets of nature, lege, et de quibusdam infiuili mysteriis. His first principle is “omnium naturae arcanorum conscios." And they are to be that all varieties of a defined locus have the same properties, so especially regarded in geometry as, by the use of " however that what is demonstrable of one should be demonstrable in like absurd expressions," classing extreme limiting forms with an manner of all, although some artifice may be required to bring infinity of intermediate cases, and placing the whole essence of a out the underlying analogy between them. The opposite thing clearly before the eyes. extremities of an infinite straight line, he says, are to be regarded Here, then, we find formulated by Kepler the doctrine of the as joined, as if the line were a circle having its centre at the concurrence of parallels at a single point at infinity and the infinity on either side of it. This leads up to the idea of a veluti principle of continuity (under the name analogy) in relation to the plus quam infinita extensio, a line-circle containing, as we say, infinitely great. Such conceptions so strikingly propounded in the line infinity. Change from the real to the imaginary state is a famous work could not escape the notice of contemporary contingent upon the passage of some element of a figure through mathematicians. Henry Briggs, in a letter to Kepler from zero or infinity and never takes place per saltum. Lines being Merton College, Oxford, dated" 10 Cal. Martiis 1625,” suggests some positive and some negative, there must be negative rectimprovements in the Ad Vitellionem paralipomena, and gives angles and negative squares, such as those of the exterior the following construction: Draw a line CBADC, and let an diameters of a hyperbola. Boscovich's first principle was that ellipse, a parabola, and a hyperbola have B and A for focus and I of Kepler, by whose quantumvis absurdis locutionibus the boldest applications of it are covered, as when we say with Poncelet | its infancy it therefore consisted of a few rules, very rough and that all concentric circles in a plane touch one another in two approximate, for computing the areas of triangles and quadri. imaginary fixed points at infinity. In G. K. Ch. von Staudt's laterals; and, with the Egyptians, it proceeded no further, the Geometrie der Lage and Beiträge zur G. der L. (Nürnberg, 1847, geometrical entities--the point, line, surface and solid-being 1856-1860) the geometry of position, including the extension of only discussed in so far as they were involved in practical affairs. the field of pure geometry to the infinite and the imaginary, is The point was realized as a mark or position, a straight line as a presented as an independent science,“ welche des Messens nicht stretched string or the tracing of a pole, a surface as an area; bedarf." (See GEOMETRY: Projective.) but these units were not abstracted; and for the Egyptians Ocular illusions due to distance, such as Roger Bacon notices geometry was only an art-an auxiliary to surveying. The in the Opus majus (i. 126, ii. 108, 497; Oxford, 1897), lead up to first step towards its elevation to the rank of a science was made or illustrate the mathematical uses of the infinite and its re- by Thales (9.0.) of Miletus, who transplanted the elementary ciprocal the infinitesimal. Specious objections can, of course, be Egyptian mensuration to Greece. Thales clearly abstracted made to the anomalies of the law of continuity, but they are the notions of points and lines, founding the geometry of the inherent in the higher geometry, which has taught us so much latter unit, and discovering per saltum many propositions conof the “ secrets of nature.” Kepler's excursus on the“ analogy” cerning areas, the circle, &c. The empirical rules of the Egyptians between the conic sections herein before referred to is given at were corrected and developed by the Ionic School which he length in an article on “ The Geometry of Kepler and Newton" | founded, especially by Anaximander and Anaxagoras, and in in vol. xviii. of the Transactions of the Cambridge Philosophical the 6th century B.C. passed into the care of the Pythagoreans. Society (1900). It had been generally overlooked, until attention From this time geometry exercised a powerful influence on was called to it by the present writer in a note read in 1880 (Proc. Greek thought. Pythagoras (9.0.), seeking the key of the C.P.S. iv. 14-17), and shortly afterwards in The Ancient and universe in arithmetic and geometry, investigated logically the Modern Geometry of Conics, with Historical Notes and Prolego- principles underlying the known propositions, and this resulted mena (Cambridge 1881). (C. T.*) in the formulation of definitions, axioms and postulates which, GEOMETRY, the general term for the branch of mathematics in addition to founding a science of geometry, permitted a which has for its province the study of the properties of crystallization, fractional, it is true, of the amorphous collection space. From experience, or possibly intuitively, we characterize of material at hand. Pythagorean geometry was essentially a existent space by certain fundamental qualities, termed axioms, geometry of areas and solids; its goal was the regular solidswhich are insusceptible of proof; and these axioms, in conjunc- the tetrahedron, cube, octahedron, dodecahedron and icosa. tion with the mathematical entities of the point, straight line, hedron-which symbolized the five elements of Greek cosmology. curve, surface and solid, appropriately defined, are the premises The geometry of the circle, previously studied in Egypt and from which the geometer draws conclusions. The geometrical much more seriously by Thales, was somewhat neglected, although axioms are merely conventions; on the one hand, the system this curve was regarded as the most perfect of all plane figures may be based upon inductions from experience, in which case and the sphere the most perfect of all solids. The circle, however, the deduced geometry may be regarded as a branch of physical was taken up by the Sophists, who made most of their discoveries science; or, on the other hand, the system may be formed by in attempts to solve the classical problems of squaring the circle, purely logical methods, in which case the geometry is a phase doubling the cube and trisecting an angle. These problems, of pure mathematics. Obviously the geometry with which we besides stimulating pure geometry, i.e. the geometry of conare most familiar is that of existent space-the three-dimensional structions made by the ruler and compasses, exercised considerspace of experience; this geometry may be termed Euclidean, able influence in other directions. The first problem led to the after its most famous expositor. But other geometries exist, discovery of the method of exhaustion for determining areas. for it is possible to frame systems of axioms which definitely Antiphon inscribed a square in a circle, and on each side an characterize some other kind of space, and from these axioms isosceles triangle having its vertex on the circle; on the sides to deduce a series of non-contradictory propositions; such of the octagon so obtained, isosceles triangles were again congeometries are called non-Euclidean. structed, the process leading to inscribed polygons of 8, 16 and It is convenient to discuss the subject matter of geometry 32 sides; and the areas of these polygons, which are easily under the following headings: determined, are successive approximations to the area of the 1. Euclidean Geometry: a discussion of the axioms of existent circle. Bryson of Heraclea took an important step when he space and of the geometrical entities, followed by a synoptical circumscribed, in addition to inscribing, polygons to a circle, account of Euclid's Elements. but he committed an error in treating the circle as the mean of II. Projective Geometry: primarily Euclidean, but differing the two polygons. The method of Antiphon, in assuming that from I. in employing the notion of geometrical continuity (9.0.)- by continued division a polygon can be constructed coincident points and lines at infinity. with the circle, demanded that magnitudes are not infinitely III. Descriplive Geometry: the methods for representing upon divisible. Much controversy ranged about this point; Aristotle planes figures placed in space of three dimensions. supported the doctrine of infinite divisibility; Zeno attempted IV. Analytical Geometry: the representation of geometrical to show its absurdity. The mechanical tracing of loci, a principle figures and their relations by algebraic equations. initiated by Archytas of Tarentum to solve the last two problems, V. Line Geomelry: an analytical treatment of the line regarded was a frequent subject for study, and several mechanical curves as the space element. were thus discovered at subsequent dates (cissoid, conchoid, VI. Non-Euclidean Geometry: a discussion of geometries quadratrix). Mention may be made of Hippocrates, who, other than that of the space of experience. besides developing the known methods, made a study of similar VII. Axioms of Geometry: a critical analysis of the foundations figures, and, as a consequence, of proportion. This step is of geometry: important as bringing into line discontinuous number and Special subjects are treated under their own headings: 6.g. continuous magnitude. PROJECTION, PERSPECTIVE;.CURVE, SURFACE; Circle, Conic SECTION; TRIANGLE, POLYGON, POLYHEDRON; there are also A fresh stimulus was given by the succeeding Platonists, who, articles on special curves and figures, e.g. Ellipse. PARABOLA, accepting in part the Pythagorean cosmology, made the study HYPERBOLA: TETRAHEDRON, CUBE,OctaHedron, DoDeCAHEDRON, of geometry preliminary to that of philosophy. The many ICOSAHEDRON;CARDIOID, CATENARY, CISSOID,Concho!D.CYCLOID, discoveries made by this school were facilitated in no small EPICYCLOID, LIMAÇON, OVAL, QUADRATRIX, SPIRAL, &c. measure by the clarification of the axioms and definitions, the History.—The origin of geometry (Gr. gn, earth, wé pov, a logical sequence of propositions which was adopted, and, more measure) is, according to Herodotus, to be found in the etymology especially, by the formulation of the analytic method, i.e. of of the word. Its birthplace was Egypt, and it arose from the assuming the truth of a proposition and then reasoning to a need of surveying the lands inundated by the Nile floods. In * For Egyptian geometry see EGYPT, $ Science and Mathematics. known truth. The main strength of the Platonist geometers | equations by intersecting conics, a step already taken by the lies in stereometry or the geometry of solids. The Pythagorcans Greeks in isolated cases, but only elevated into a method by Omar had dealt with the sphere and regular solids, but the pyramid, al Hayyami, who flourished in the inth century. During the prism, cone and cylinder were but little known until the Platonists middle ages little was added to Greek and Arabic geometry. took them in hand. Eudoxus established their mensuration, Leonardo of Pisa wrote a Practica geometriae (1220), wherein proving the pyramid and cone to have one-third the content Euclidean methods are employed; but it was not until the 14th of a prism and cylinder on the same base and of the same height, century that geometry, generally Euclid's Elements, became and was probably the discoverer of a proof that the volumes of an essential item in university curricula. There was, however, spheres are as the cubes of their radii. The discussion of sections no sign of original development, other branches of mathematics, of the cone and cylinder led to the discovery of the three curves mainly algebra and trigonometry, exercising a greater fascination named the parabola, ellipse and hyperbola (see CONIC SECTION); until the 16th century, when the subject again came into favour. it is difficult to over-estimate the importance of this discovery; The extraordinary mathematical talent which came into being its investigation marks the crowning achievement of Greek in the 16th and 17th centuries reacted on geometry and gave rise geometry, and led in later years to the fundamental theorems to all those characters which distinguish modern from ancient and methods of modern geometry. geometry. The first innovation of moment was the formulation The presentation of the subject matter of geometry as a con of the principle of geometrical continuity by Kepler. The notion nected and logical series of propositions, prefaced by "Opol or of infinity which it involved permitted generalizations and foundations, had been attempted by many; but it is to Euclid systematizations hitherto unthought of (see GEOMETRICAL that we owe a complete exposition. Little indeed in the Elements CONTINUITY); and the method of indefinite division applicd to is probably original except the arrangement; but in this Euclid rectification, and quadrature and cubature problems avoided surpassed such predecessors as Hippocrates, Leon, pupil of the cumbrous method of exhaustion and provided more accurate Neocleides, and Theudius of Magnesia, devising an apt logical results. Further progress was made by Bonaventura Cavalieri, model, although when scrutinized in the light of modern mathe- who, in his Geometria indivisibilibus continuorum" (1620), de. matical conceptions the proofs are riddled with fallacies. Accordvised a method intermediate between that of exhaustion and ing to the commentator Proclus, the Elements were written with the infinitesimal calculus of Leibnitz and Newton. The logical a twofold object, first, to introduce the novice to geometry, and basis of his system was corrected by Roberval and Pascal; and secondly, to lead him to the regular solids; conic sections found their discoveries, taken in conjunction with those of Leibnitz, no place therein. What Euclid did for the line and circle, Newton, and many others in the fluxional calculus, culminated Apollonius did for the conic sections, but there we have a discoverer in the branch of our subject known as differential geometry as well as editor. These two works, which contain the greatest (see INFINITESIMAL CALCULUS, CURVE; SURFACE). contributions to ancient geometry, are treated in detail in A second important advance followed the recognition that Section I. Euclidean Geometry and the articles EUCLID; CONIC conics could be regarded as projections of a circle, a conception SECTION; APOLLONIUS. Between Euclid and Apollonius there which led at the hands of Desargues and Pascal to modern flourished the illustrious Archimedes, whose geometrical dis- projective geomelry and perspeclive. A third, and perhaps the coveries are mainly concerned with the mensuration of the most important, advance attended the application of algebra to circle and conic sections, and of the sphere, cone and cylinder, geometry by Descartes, who thereby founded analylical geometry. and whose greatest contribution to geometrical method is the The new fields thus opened up were diligently explored, but the elevation of the method of exhaustion to the dignity of an instru-calculus exercised the greatest attraction and relatively little ment of research. Apollonius was followed by Nicomedes, the progress was made in geometry until the beginning of the 19th inventor of the conchoid; Diocles, the inventor of the cissoid; century, when a new era opened. Zenodorus, the founder of the study of isoperimetrical figures; Gaspard Monge was the first important contributor, stimulating Hipparchus, the founder of trigonometry; and Heron the elder, analytical and differential geometry and founding descriplice who wrote after the manner of the Egyptians, and primarily geometry in a series of papers and especially in his lectures at the directed attention to problems of practical surveying. Ecole polytechnique. Projective geometry, founded by DesarOf the many isolated discoveries made by the later Alexandrian gues, Pascal, Monge and L. N. M. Carnot, was crystallized by mathematicians, those of Menelaus are of importance. He J. V. Poncelet, the creator of the modern methods. In his showed how to treat spherical triangles, establishing their Traité des propriétés des figures (1822) the line and circular points properties and determining their congruence; his theorem on at infinity, imaginaries, polar reciprocation, homology, crossthe products of the segments in which the sides of a triangle ratio and projection are systematically employed. In Germany, are cut by a line was the foundation on which Carnot erected A.F. Möbius, J. Plücker and J. Steiner were making far-reaching bis theory of transversals. These propositions, and also those contributions. Möbius, in his Barycentrische Calcri (1827), of Hipparchus, were utilized and developed by Ptolemy (q.v.), introduced homogeneous co-ordinates, and also the powerful the expositor of trigonometry and discoverer of many isolated notion of geometrical transformation, including the special propositions. Mention may be made of the commentator Pappus, cases of collineation and duality; Plücker, in his Analytischwhose Mathematical Collections is valuable for its wealth of geometrische Entwickelungen (1828-1831), and his System der historical matter; of Theon, an editor of Euclid's Elements and analytischen Gcomelrie (1835), introduced the abridged notation, commentator of Ptolemy's Almagest; of Proclus, a commentator line and plane co-ordinates, and the conception of generalized of Euclid; and of Eutocius, a commentator of Apollonius and space elements; while Steiner, besides enriching geometry in Archimedes. numerous directions, was the first to systematically generate The Romans, essentially practical and having no inclination figures by projective pencils. We may also notice M. Chasles, to study science qua science, only bad a geometry which sufficed whose A pergu historique (1837) is a classic. Synthetic geometry, for surveying; and even here there were abundant inaccuracies, characterized by its fruitfulness and beauty, attracted most the empirical rules employed being akin to those of the Egyptians attention, and it so happened that its originally weak logical and Heron. The Hindus, likewise, gave more attention to foundations became replaced by a more substantial set of axioms. computation, and their geometry was either of Greek origin or These were found in the anharmonic ratio, a device leading to in the form presented in trigonometry, more particularly con- the liberation of synthetic geometry from metrical relations, nected with arithmetic. It had no logical foundations; each and in involution, which yielded rigorous definitions of imaginproposition stood alone; and the results were empirical. The aries. These innovations were made by K. J. C. von Staudt. Arabs more closely followed the Greeks, a plan adopted as a Analytical geometry was stimulated by the algebra of invariants, sequel to the translation of the works of Euclid, Apollonius, a subject much developed by A. Cayley, G. Salmon, S. H. AronArchimedes and many others into Arabic. Their chief con- hold, L. O. Hesse, and more particularly by R. F. A. Clebsch. tribution to geometry is exhibited in their solution of algebraic The introduction of the line as a space element, initiated by H. Grassmann (1844) and Cayley (1859), yielded at the hands of Def. 3, I.,“ The extremities of a line are points," is a proposition Plücker a new geometry, termed line geometry, a subject which either has to be proved, and then it is a theorem, or which developed more notably by F. Klein, Clebsch, C. T. Reye and has to be taken for granted, in which case it is an axiom. And F.O. R. Sturm (see Section V., Line Geometry). so with Def. 6, I., and Def. 2, XI. Non-euclidean geometries, having primarily their origin in the $ 3. Euclid's definitions mentioned above are attempts to discussion of Euclidean parallels, and treated by Wallis, Saccheri describe, in a few words, notions which we have obtained by and Lambert, have been especially developed during the 19th inspection of and abstraction from solids. A few more notions century. Four lines of investigation may be distinguished:- have to be added to these, principally those of the simplest the naive-synthetic, associated with Lobatschewski, Bolyai, line-the straight line, and of the simplest surface-the flat Gauss; the metric differential, studied by Riemann, Helmholtz, surface or plane. These notions we possess, but to define them Beltrami; the projective, developed by Cayley, Klein, Clifford; accurately is difficult. Euclid's Definition 4, I., "A straight and the critical-synthetic, promoted chiefly by the Italian line is that which lies evenly between its extreme points," must mathematicians Peano, Veronese, Burali-Forte, Levi Civittà, be meaningless to any one who has not the notion of straightness and the Germans Pasch and Hilbert. (C. E.*) in his mind. Neither does it state a property of the straight line which can be used in any further investigation. Such a I. EUCLIDEAN GEOMETRY property is given in Axiom 10, I. It is really this axiom, together This branch of the science of geometry is so named since its with Postulates 2 and 3, which characterizes the straight line. methods and arrangement are those laid down in Euclid's Whilst for the straight line the verbal definition and axiom Elements. are kept apart, Euclid mixes them up in the case of the plane. § 1. Axioms.—The object of geometry is to investigate the Here the Definition 7, I., includes an axiom. It defines a plane properties of space. The first step must consist in establishing as a surface which has the property that every straight line ihose fundamental properties from which all others follow by which joins any two points in it lies altogether in the surface. processes of deductive reasoning. They are laid down in the But if we take a straight line and a point in such a surface, and Axioms, and these ought to form such a system that nothing draw all straight lines which join the latter to all points in the need be added to them in order fully to characterize space, and first line, the surface will be fully determined. This construction that nothing may be omitted without making the system in- is therefore sufficient as a definition. That every other straight complete. They must, in fact, completely " define "space. line which joins any two points in this surface lies altogether 82. Definitions. The axioms of Euclidean Geometry are in it is a further property, and to assume it gives another axiom. obtained from inspection of existent space and of solids in Thus a number of Euclid's axioms are hidden among his first existent space,-hence from experience. The same source definitions. A still greater confusion exists in the present gives us the notions of the geometrical entities to which the editions of Euclid between the postulates and axioms so called, axioms relate, viz. solids, surfaces, lines or curves, and points. but this is due to later editors and not to Euclid himself. The A solid is directly given by experience; we have only to abstract latter had the last three axioms put together with the postulates all material from it in order to gain the notion of a geometrical (airhuara), so that these were meant to include all assumptions solid. This has shape, size, position, and may be moved. Its relating to space. The remaining assumptions, which relate to boundary or boundaries are called surfaces. They separate one magnitudes in general, viz. the first eight“ axioms " in modern part of space from another, and are said to have no thickness. editions, were called common notions” (kouval évvocal). Their boundaries are curves or lines, and these have length of the latter a few may be said to be definitions. Thus the eighth only. Their boundaries, again, are points, which have no might be taken as a definition of "equal,” and the seventh magnitude but only position. We thus come in three steps of "halves.” If we wish to collect the axioms used in Euclid's from solids to points which have no magnitude; in each step Elements, we have therefore to take the three postulates, the we lose one extension. Hence we say a solid has three dimensions, last three axioms as generally given, a few axioms hidden in the a surface two, a line one, and a point none. Space itself, of which definitions, and an axiom used by Euclid in the proof of Prop, a solid forms only a part, is also said to be of three dimensions. 4, I, and on a few other occasions, viz. that figures may be The same thing is intended to be expressed by saying that a moved in space without change of shape or size. solid has length, breadth and thickness, a surface length and $ 4. Postulates.--The assumptions actually made by Euclid breadth, a line length only, and a point no extension whatsoever. may be stated as follows: Euclid gives the essence of these statements as definitions: (1) Straight lines exist which have the property that any one of Def. 1, 1. A point is thal which has no parts, or which has no mag them may be produced both ways without limit, that through any nitude. two points in space such a line may be drawn, and that any two of Def. 2, I. A line is length without breadtk. them coincide throughout their indefinite extensions as soon as two Def. 5. I. A superficies is that which has only length and breadth. points in the one coincide with two points in the other. (This Def. 1, XI. A solid is that which has length, breadth and thickness. gives the contents of Def. 4, part of Def. 35, the first two Postulates, It is to be noted that the synthetic method is adopted by and Axiom 10.) (2) Plane surfaces or planes exist having the property laid down Euclid; the analytical derivation of the successive ideas of in Del. 7, that every straight line joining any two points in such a "surface," "line,” and “point" from the experimental realiza- surface lies altogether in it. tion of a “solid” does not find a place in his system, although (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, possessing more advantages. and wherever in that plane we construct a right angle, all angles If we allow motion in geometry, we may generate these thus constructed will be equal, so that any one of them may be made entities by moving a point, a line, or a surface, thus: to coincide with any other. (Axiom 11.) The path of a moving point is a line. (4) The 12th Axiom of Euclid. This we shall not state now, but The path of a moving line is, in general, a surface. only introduce it when we cannot proceed any further without it. The path of a moving surface is, in general, a solid. (5) Figures may be freely moved in space without change of shape or size. This is assumed by Euclid, but not stated as an axiom. And we may then assume that the lines, surfaces and solids, (6) In any plane a circle may be described, having any point in as defined before, can all be generated in this manner. From that plane as centre, and its distance from any other point in that this generation of the entities it follows again that the boundaries plane as radius. (Postulate 3.) .-the first and last position of the moving element-of a line are The definitions which have not been mentioned are all points, and so on; and thus we come back to the considerations “ nominal definitions," that is to say, they fix a name for a with which we started. thing described. Many of them overdetermine a figure. Euclid points this out in his definitions,-Def. 3, I., Def. 6, I., & S. Euclid's Elements (see EUCLID) are contained in thirteen and Def. 2, XI. He does not, however, show the connexion books. Of these the first four and the sixth are devoted to which these definitions have with those mentioned before. plane geometry," as the investigation of figures in a plane is When points and lines have been defined, a statement like I generally called. The sth book contains the theory of proportion |