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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 See also:

report of the See also:Institute prefixed to See also:Jean See also:Victor See also:Poncelet's Traite See also:des preprieies projectives des figures (See also:Paris, 1822), it is said that he employed " ce qu'il appelle le principe de continuity." The See also:law or principle thus named by him had, he tells us, been tacitly assumed as axiomatic by " See also:les plus savans geometres." It had in fact been enunciated as " lex continuationis,',' and" la loi de la continuity," by Gottfried Wilhelm See also:Leibnitz (Oaf. N.ED.), and previously under another name by Johann See also:Kepler in cap. iv. 4 of his Ad Vitellionem paralipomena quibus astronomiae pars optica traditur (Francofurti, 1604). Of sections of the See also:cone, he says, there are five See also:species from the " recta linea " or See also:line-pair to the circle. From the line-pair we pass through an infinity of hyperbolas to the See also: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 See also:focus at the centre, an See also:ellipse or See also: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 See also:axis wit/See also:tin or without the See also:curve. The line from it to any point of the See also:section is parallel to the axis. To carry out the See also:analogy we must speak paradoxically, and say that the line-pair likewise has foci, which in this See also: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 See also:

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 See also:doctrine of the concurrence of See also:parallels at a single point at infinity and the principle of continuity (under the name analogy) in relation to the, infinitely See also:great. Such conceptions so strikingly propounded in a famous See also:work could not See also:escape the. See also:notice of contemporary mathematicians. See also:Henry See also:Briggs, in a See also:letter to Kepler from Merton See also:College, See also: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 See also: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 See also:call the directrix, this taking the See also: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 See also:left no See also:record, had already taken up the new doctrine in geometry in its author's lifetime. Six years after Kepler's See also:death in 163o See also:Girard Desargues, " the Nlonge of his See also:age," brought out the first of his remarkable See also:works founded on the same principles, a See also:short See also:tract entitled tlTethode universelle de meltre en See also:perspective les objets See also:donne; reellement ou en devis (Paris, 1636); but " Le See also:privilege etoit de 163o" (Poudra, Euvres de Des., i.

55). Kepler as a See also:

modern geometer is best known by his New Stereometry of See also:Wine Casks (Lincii,1615), in which he replaces the circuitous Archimedeau method of exhaustion by a See also:direct " royal road " of infinitesimals, treating a vanishing arc as a straight line and regarding a curve as made up of a See also:succession of short chords. Some 2000 years previously one Antipho, probably the well-known opponent of See also:Socrates, has regarded a circle in like manner as the limiting See also:form of a many-sided inscribed rectilinear figure. Antipho's notion was rejected by the men of his See also: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 See also:reason, for it rested on an assumed law of continuity rather than on palpable See also:proof. To See also:complete the theory of continuity, the one thing needful was the See also:idea of imaginary points implied in the algebraical geometry of Rene See also:Descartes, in which equations between variables representing co-ordinates were found often to have imaginary roots. See also:Newton, in his two sections on " Inventio orbium (Principia i. 4, 5), shows in his brief way that he is See also:familiar with the principles of modern geometry. In two propositions he uses. an See also: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 See also: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 See also: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 See also:

state in such manner as not to vitiate conclusions arrived at on the See also:hypothesis of their reality. See also:Roger See also:Joseph See also:Boscovich, a careful student of Newton's works, has a full and thorough discussion of geometrical continuity in the third and last See also:volume of his Eleinenta universae inatheseos (ed. See also: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 See also: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 See also: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 See also: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. See also:Change from the real to the imaginary state is *contingent upon the passage of some See also:element of a figure through zero or infinity and never takes place per saltum. Lines being some See also: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 See also: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, See also:surface and solid—being only discussed in so far as they were involved in See also:practical affairs. The point was realized as a See also:mark or position, a straight line as a stretched See also:string or the tracing of a See also:pole, a surface as an See also:area; but these See also:units were not abstracted; and for the Egyptians geometry was only an art—an auxiliary to See also:surveying.) The first step towards its See also:elevation to the See also:rank of a See also:science was made by Thales (q.v.) of See also:Miletus, who transplanted the elementary See also:Egyptian See also:mensuration to See also:Greece. Thales clearly abstracted the notions of points and lines, See also: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 See also:developed by the Ionic School which he founded, especially by Anaximander and Anaxagoras, and in the 6th See also:century B.C. passed into the care of the Pythagoreans.

From this See also:

time geometry exercised a powerful See also:influence on See also:Greek thought. See also:Pythagoras (q.v.), seeking the See also:key of the universe in See also:arithmetic and geometry, investigated logically the principles underlying the known propositions; and this resulted in the formulation of See also:definitions, axioms and postulates which, in addition to founding a science of geometry, permitted a See also:crystallization, fractional, it is true, of the amorphous collection of material at See also:hand. See also:Pythagorean geometry was essentially a geometry of areas and solids; its See also:goal was the See also:regular solids—the See also:tetrahedron, See also:cube, See also:octahedron, See also:dodecahedron and,icosahedron—which symbolized the five elements of Greek cosmology. The geometry of the circle,. previously studied in See also:Egypt and much more seriously by Thales, was somewhat neglected, although this curve was regarded as the most perfect of all See also:plane figures and the See also:sphere the most perfect of all solids. The circle, however, was taken up by the See also:Sophists, who made most of their discoveries in attempts to solve the classical problems of squaring the circle, doubling the cube and trisecting an See also: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 See also:discovery of the method of exhaustion for determining areas. See also: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 See also: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 See also:Heraclea took an important step when he circumscribed, in addition to inscribing, polygons to a circle, but he committed an See also:error in treating the circle as the mean of the two polygons. The method of Antiphon, in assuming that by continued See also:division a See also:polygon can be constructed coincident with the circle, demanded that magnitudes are not infinitely divisible. Much controversy ranged about this point; See also:Aristotle supported the doctrine of infinite divisibility; See also:Zeno attempted to show its absurdity. The See also:mechanical tracing of loci, a principle initiated by See also:Archytas of Tarenturn to solve the last two problems, was a frequent subject for study, and several mechanical curves were thus discovered at subsequent See also:dates (See also:cissoid, See also:conchoid, See also:quadratrix).

Mention may be made of See also:

Hippocrates, who, besides developing the known methods, made a study of similar ,A fresh stimulus was given by. the succeeding Platonists, who, accepting in See also:part the Pythagorean cosmology, made the study of geometry preliminary to that of See also: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 See also:analytic method, i,e. of assuming the truth of a proposition and then reasoning to a I For Egyptian geometry see EGYPT. § Science and See also:Mathematics. applications of it are covered, as when we say with Poncelet ' that all concentric circles in a plane See also: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 See also:extension of the See also:field of pure geometry to the infinite and the imaginary, is presented as an See also:independent science, " welche des Messens nicht bedarf." (See GEOMETRY: Projective.) Ocular illusions due to distance, such as Roger See also:Bacon notices in the See also:Opus majus (i. 126, ii. 1o8, 497; Oxford, 1897), See also: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 See also:article on " The Geometry of Kepler and Newton " in vol. xviii. of the Transactions of the See also:Cambridge Philosophical Society (19oo). It had been generally overlooked, until See also:attention was called to it by the See also:present writer in a See also:note read in 188o (Prot.

C.P.S. iv. 14–17), and shortly afterwards in The See also:

Ancient and Modern Geometry of Conics, with See also:Historical Notes and Prolegomena (Cambridge 1881), (C.

End of Article: GEOMETRICAL CONTINUITY

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