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GEOMETRICAL CONTINUITY

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Originally appearing in Volume V11, Page 675 of the 1911 Encyclopedia Britannica.
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GEOMETRICAL CONTINUITY. In a report of the Institute prefixed to Jean Victor Poncelet's Traite des preprieies projectives des figures (Paris, 1822), it is said that he employed " ce qu'il appelle le principe de continuity." The law or principle thus named by him had, he tells us, been tacitly assumed as axiomatic by " les plus savans geometres." It had in fact been enunciated as " lex continuationis,',' and" la loi de la continuity," by Gottfried Wilhelm Leibnitz (Oaf. N.ED.), 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 wit/tin 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, " omniunt uaturae 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 paralipometta, and gives the following construction: Draw a line CBADC, and let an ellipse, a parabola, and a, hyperbola have B and A for focus andvertex. 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 in-finitely distant, is equidistant from the focus B and the line through D which we call the directrix, this taking the place of either circle when its centre C is at infinity, and every line CP being then parallel to the axis. Thus Briggs, and we know not how many " savans geometres " who have left no record, had already taken up the new doctrine in geometry in its author's lifetime. Six years after Kepler's death in 163o Girard Desargues, " the Nlonge of his age," brought out the first of his remarkable works founded on the same principles, a short tract entitled tlTethode universelle de meltre en perspective les objets donne; reellement ou en devis (Paris, 1636); but " Le privilege etoit de 163o" (Poudra, Euvres 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 Archimedeau 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 with-out reason, for it rested on an assumed law of continuity rather than on palpable proof. To complete the theory of continuity, the one thing needful was the idea of imaginary points implied in the algebraical geometry of Rene Descartes, in which equations between variables representing co-ordinates were found often to have imaginary roots. Newton, in his two sections on " Inventio orbium (Principia i. 4, 5), shows in his brief way that he is familiar with the principles of modern geometry. In two propositions he uses. an auxiliary line which is supposed to cut the conic in X and Y, but, as he remarks at the end of the second (prop. 24), it may not cut it at all. For the sake of brevity he passes on at once with the observation that the required constructions are evident from the case in which the line cuts the trajectory. In the scholium appended to prop. 27, after saying that an asymptote is a tangent at infinity, he gives an unexplained general construction for the axes of a conic, which seems to imply that it has asymptotes. In all such cases, having equations to his loci in the background, he may have thought of elements of the figure as passing into the imaginary state in such manner as not to vitiate conclusions arrived at on the hypothesis of their reality. Roger Joseph Boscovich, a careful student of Newton's works, has a full and thorough discussion of geometrical continuity in the third and last volume of his Eleinenta universae inatheseos (ed. prim. Venet, 1757), which contains Sectionum conicarum elementa nova quadam methodo concinnata at dissertationem de transformatione locorum geometricorurn, ubi de continuitatis lege, et de quibusdam irtfinili 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 quoit infinita, e.xtensio, 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 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.) The first step towards its elevation to the rank of a science was made by Thales (q.v.) of Miletus, who transplanted the elementary Egyptian mensuration to Greece. Thales clearly abstracted the notions of points and lines, founding the geometry of the latter unit, and discovering per saltum many propositions concerning areas, the circle, &c. The empirical rules of the Egyptians were corrected and developed by the Ionic School which he founded, especially by Anaximander and Anaxagoras, and in the 6th century B.C. passed into the care of the Pythagoreans. From this time geometry exercised a powerful influence on Greek thought. Pythagoras (q.v.), seeking the key of the universe in arithmetic and geometry, investigated logically the principles underlying the known propositions; and this resulted in the formulation of definitions, axioms and postulates which, in addition to founding a science of geometry, permitted a crystallization, fractional, it is true, of the amorphous collection of material at hand. Pythagorean geometry was essentially a geometry of areas and solids; its goal was the regular solids—the tetrahedron, cube, octahedron, dodecahedron and,icosahedron—which symbolized the five elements of Greek cosmology. The geometry of the circle,. previously studied in Egypt and much more seriously by Thales, was somewhat neglected, although this curve was regarded as the most perfect of all plane figures and the sphere the most perfect of all solids. The circle, however, was taken up by the Sophists, who made most of their discoveries in attempts to solve the classical problems of squaring the circle, doubling the cube and trisecting an angle. These problems, besides stimulating pure geometry, i.e. the geometry of constructions made by the ruler and compasses, exercised consider-able influence in other directions. The first problem led to the discovery of the method of exhaustion for determining areas. Antiphon inscribed a square in a circle, and on each side an isosceles triangle having its vertex on the circle; on the sides of the octagon so obtained, isosceles triangles were again constructed, the process leading to inscribed polygons of 8, 16 and 32 sides; and the areas of these polygons, which are easily determined, are, successive approximations to the area of the circle. Bryson of Heraclea took an important step when he circumscribed, in addition to inscribing, polygons to a circle, but he committed an error in treating the circle as the mean of the two polygons. The method of Antiphon, in assuming that by continued division a polygon can be constructed coincident with the circle, demanded that magnitudes are not infinitely divisible. Much controversy ranged about this point; Aristotle supported the doctrine of infinite divisibility; Zeno attempted to show its absurdity. The mechanical tracing of loci, a principle initiated by Archytas of Tarenturn 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 ,A fresh stimulus was given by. the succeeding Platonists, who, accepting in part the Pythagorean cosmology, made the study of geometry preliminary to that of philosophy. The many discoveries made by this school were facilitated in no small measure by the clarification of the axioms and definitions, the logical sequence of propositions which was adopted, and, more especially, by the formulation of the analytic method, i,e. of assuming the truth of a proposition and then reasoning to a I For Egyptian geometry see EGYPT. § Science and Mathematics. applications of it are covered, as when we say with Poncelet ' that all concentric circles in a plane touch one another in two imaginary fixed points at infinity. In G. K. Ch. von Staudt's Geometric der Loge and Beitrage zur G. der L. (Ni.irnberg, 1847, 1856–186o) the geometry of position, including the extension of the field of pure geometry to the infinite and the imaginary, is presented as an independent science, " welche des Messens nicht bedarf." (See GEOMETRY: Projective.) Ocular illusions due to distance, such as Roger Bacon notices in the Opus majus (i. 126, ii. 1o8, 497; Oxford, 1897), lead up to or illustrate the mathematical uses of the infinite and its reciprocal the infinitesimal. Specious objections can, of course, be made to the anomalies of the law of continuity, but they are inherent in the higher geometry, which has taught us so much of the " secrets of nature." Kepler's excursus on the " analogy " between the conic sections hereinbefore referred to is given at length in an article on " The Geometry of Kepler and Newton " in vol. xviii. of the Transactions of the Cambridge Philosophical Society (19oo). It had been generally overlooked, until attention was called to it by the present writer in a note read in 188o (Prot. C.P.S. iv. 14–17), and shortly afterwards in The Ancient and Modern Geometry of Conics, with Historical Notes and Prolegomena (Cambridge 1881), (C. T.*)
End of Article: GEOMETRICAL CONTINUITY
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