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 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’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 Traité de géologie (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, 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’s Text Book of Geology. See also the separate articles on geological subjects for special references to authorities.
GEOMETRICAL CONTINUITY. In a report of the Institute prefixed to Jean Victor Poncelet’s Traité 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 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 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 entitled Méthode 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 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 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” (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
