Page:The New International Encyclopædia 1st ed. v. 16.djvu/209

POLE. was interrupted by the death of the Queen, November 17, 1558. Pole died within twelve hours afterwards. Besides the treatise De Unitate, already mentioned, he is also the author of a book De Concilio, and of other treatises on the authority of the Roman pontiff and the reformation of England, and of many important letters, full of interest for the history of the time. Consult his Life by A. Zimmermann (Regensburg, 1893); also a study of the first and last parts of his life by F. G. Lee (London, 1887).

POLE, WILLIAM (1814-1900). An English engineer, musician, and writer on whist. He was born at Birmingham, studied engineering there, became professor of engineering at Elphinston College, Bombay (1844), and in 1850 was employed in calculations for the Menai Bridge. After acting as assistant to James Meadows Rendel and to Sir John Fowler, he entered business independently in 1858, and in 1871 was made consulting engineer of the Japanese Government. He was a prominent musician, being musical examiner in London University from 1878 to 1891, and as an authority on whist ranks with Henry Jones. Pole's works are: On the Strength and Defects of Beams (1850); Iron as a Material of Construction (1872); Life of Siemens (1888); The Theory of the Modern Scientific Game of Whist (1865, and often); The Philosophy of Whist (1883 and often); The Evolution of Whist (1895); The Philosophy of Music (1877; 4th ed., 1895), and Mozart's Requiem (1879).

POLE, WILLIAM. Jr. (1852—). An English playwright, born in London. Turning his attention to the stage in lS7(i, he became manager of Royal Victoria Hall, London (1881-83). and stage manager of F. R. Benson's company (1883-84). In 1895 he founded the Elizabethan Stage Society, a natural outcome of his endeavors to revive interest in the Elizabethan stage and drama. Under his management have been produced Hamlet, without scenery (Saint George's Hall, 1881); Webster's Duchess of Malfi (Independent Theatre, 1892); Measure for Measure (Royalty Theatre, 1893); Marlowe's Dr. Faustus; Arden of Feversham; Ford's Hrolcen Heart; Ben Jonson's Sad Shepherd and Alchemist; Milton's Samson Agonistes; Fitzgerald's translation of Calderon's Life's a Dream; Kalidasa's Siikiintalu: Coleridge's translation of Schiller's Wallenstein; and other old plays. In 1884 he dramatized Howells's .1 Forrr/one Conclusion for the 01ym|iic Theatre: and in 1886, Baring-fiould's Mehaluh for the Gaiety Theatre.

POLE AND POLAR. The secant drawn through the points of contact of two intersecting tangents to a conic is called the polar of the point of intersection with respect to the conic, and the point is called the pole of the secant. Any secant of a conic through a given point is cut harmonically by the curve and the polar of O. Two points are said to be conjugate with respect to a conic when each lies on the polar of the other. Two straight lines are said to be conjugate with respect to a conic when each passes through the pole of the other. Thus conjugate diameters are conjugate lines through the centre. When the pole is inside the conic, the tangents are imaginary, and the polar line fails to cut the conic in real points. In this case the locus of the harmonic conjugate of the pole serves ns a more suggestive definition of the polar. If the pole is on the conic, the polar becomes a tangent at the pole. The polar of the focus is the directri. in the ellipse, hyperbola, and parabola (qq.v.). A few of the "relations which give remarkable power to the theory of polars in the donuiin of geometry are: (l)"The polars of collinear points with respect to a conic are a pencil of lines passing through the pole of the line, and conversely. (2) If the vertices of a triangle are the poles of the sides of another triangle, the vertices of the latter are the poles of the sides of the former. Such triangles are said to be conjugate to each other with respect to the conic. (3) If the sides of a triangle are the polars of its own vertices, the triangle is called a self-conjugate triangle. (4) The poles and polars of the lines and points of rectilinear plane figures with respect to a co-planar conic form a rectilinear figure called the polar reciprocal of the given figure with respect to the auxiliary conic. The method of reciprocal polars obtains from any given theorem concerning the positions of points and lines another theorem in which straight lines take the place of points, and points of straight lines. (See Duality.) Thus a line joining two points in one figure corresponds to a point determined by two intersecting lines in the reciprocal figure. Since the pole of any line through the centre of the auxiliary conic is at infinity, the points at infinity on the reciprocal curve corrcsjKind to the tangents to the original curve from the centre of the auxiliary conic. Mence the reciprocal of a conic is an hyperbola, narabola. or ellipse, according as the tangents to it through the centre of the auxiliary conic are real, coincident, or imaginary. Pascal's and Brianchon's theorems are reciprocals. (See Concurrence and CoLLiNEARiTY.) Conjugate lines or conjugate points project into conjugate lines or points (see Projection), hence the relations of pole and polar are unaltered by projection. The relations between pole and polar were known to the ancients, but Desargues (1639) was the first to develop the theory. To Servois ( 1810) Is due the name pole (in this sen.se) and to Gergonne (1812) the name polar. Steiner (1848) also treated the subject exhaustively. Hesse (1837,1842) introduced the notion of polar triangles, polar tetrahedra, and systems of conjugate points as the geometric expressions of analytic relations. The terms 'pole' and 'polar' have other meanings in mathematics than those already mentioned. The centre from which radii vectores are drawn in a system of polar coiirdinates (see CoiiRiirNATES) is called the pole. The extremities of the diameter of a sphere, perpendicular to one of its circles, are called poles of the circle. If the sides of a spherical triangle ai-e arcs of circles, whose poles are the vertices of another triangle, the latter is called the polar triangle of the first. It is shown in geometry that if one triangle is the polar of another, the second is a polar of the first. For the bibliography of the subject of polars of various orders, consult .Toachimsthal in the yo'.trelles Annalcs dc Mathemati<irie^' (1850); fnterm^diarc des Mathematicirns (1890). F(U- the theory of the subject, consult Reye, Oeometrir der Laac (Leipzig. 2d ed., 188286; English ed.. New York, 1898); Salmon. Higher Plane Curre.i (Dublin. 1852).

POLEBURN. A disease of tobacco (q.v.).