EUCLIDEAN GEOMETRY

study of points, lines, angles, surfaces, and solids on the basis of the 10 axioms and postulates selected by the Greek mathematician Euclid (c. 300 BC). One of the outstanding achievements of the ancient Greeks was the construction of a deductive system of geometry, which, beginning with principles that they regarded as obviously true and derived from experience, culminated in quite deep theorems, some of which are still an important part of mathematics. The elementary part of the deductive system of geometry was set forth in Euclid's Elements, and, until the early 20th century, Euclidean geometry meant the material in that book and others written in the same spirit. Euclid reasoned on the figure drawn on the page or envisioned in the imagination, and he often assumed details and relations read from the figure that were not explicitly stated. The figure was an important constituent of the proof, although it was never made clear how far it should be used. Partly because Euclid's geometry was considered to be the only geometry, no one bothered to question or examine many of its details. Toward the end of the 19th century another view emerged, and Euclidean geometry is now regarded as merely one example of an abstract mathematical doctrine. In any deductive theory, because each theorem is proved from preceding theorems, a beginning must be made somewhere with unproved assumptions. These assumptions are called axioms, replacing Euclid's word postulates. Whether, or in what sense, they are true is no concern of the theory: any set of axioms may be laid down provided that they do not yield two contradictory theorems. Likewise, as each technical term (e.g., square or perpendicular) is defined by reference to earlier terms, this chain cannot lead backward indefinitely, and a beginning must be made with certain undefined terms. What these terms mean is of no concern in the abstract theory, although they may acquire meaning in some application of the theory. The undefined terms may be elements, such as point, line, or relations such as lies on in the statement: the point A lies on the line l. If, for example, a triangle ABC is defined as the set of points on the sides BC, CA, AB, together with the points A, B, C, then the terms point and side must either have been defined previously or be taken as undefined. The term set, however, is a concept of logic, and it is assumed that logic has already been developed. Many writers on an abstract doctrine try to reduce the number of undefined terms and of axioms to a minimum. Until the 20th century the book of Euclid or one of its many variants was used in all schools, a use, however, that was ultimately attacked justifiably from two sides: Euclid was too abstract for beginners, yet not exact enough to satisfy modern requirements. The modern abstract treatment was initiated by mathematicians in Germany and Italy, by Moritz Pasch in 1882, Giuseppe Peano in 1889 and Mario Pieri. The most influential work was that by the well-known German mathematician David Hilbert, Grundlagen der Geometrie (1899; The Foundations of Geometry, 1902). Euclid, who lived in Alexandria in the time of the first Ptolemy (323285/283 BC), systematized in his book the work of his predecessors, beginning with Pythagoras (died c. 500 BC) and his followers. It was the basis of the more advanced studies of Apollonius of Perga (3rd century BC) on conics and of Archimedes (3rd century BC) on mechanics and the areas of circles. The Elements was translated into Arabic during the reign of Harun ar-Rashid (786809), and the first Latin version in a complete form was made from the Arabic by the English scholastic philosopher Adelard of Bath about 1120. The standard English edition is now that of Sir Thomas Little Heath, published in 1908. Additional reading Euclid, The Thirteen Books of Euclid's Elements, 2nd ed. rev., 3 vol. (1926, reissued 1956), is the standard English translation, with extensive commentary by Thomas L. Heath. David Hilbert, Foundations of Geometry, 2nd ed. (1971, reissued 1992; originally published in German, 10th ed., rev. and enlarged, 1968), gives a logical account of Euclid's own methods and many new results. The axiomatic approach now common in all branches of mathematics is due to the influence of this book and to the work of Giuseppe Peano and his school. Henry G. Forder, Foundations of Euclidean Geometry (1927, reprinted 1958), expounds in full detail the work of Hilbert, his followers, and the American and Italian schools. H.S.M. Coxeter and S.L. Greitzer, Geometry Revisited (1967), is a pleasant account of advanced theorems in Euclidean geometry. Studies of Euclidean geometry in its setting in general geometry include Henry G. Forder, Geometry, 2nd ed. (1960); H.S.M. Coxeter, Introduction to Geometry, 2nd ed. (1969, reissued 1989); Daniel Pedoe, A Course of Geometry for Colleges and Universities (1970, reprinted as Geometry: A Comprehensive Course, 1988); Judith N. Cederberg, A Course in Modern Geometries (1989); and Henry Parker Manning, Geometry of Four Dimensions (1914, reprinted 1956), an excellent work. A recent survey of work in some branches of geometry (e.g., convexity, computational geometry, and curvature) is in a special issue of The American Mathematical Monthly, vol. 97, no. 8 (October 1990), devoted to geometry. Some advanced topics are presented in Alfred S. Posamentier, Excursions in Advanced Euclidean Geometry, rev. ed. (1984). I.M. Yaglom (I.M. Iaglom), Geometric Transformations, 3 vol. (196273; originally published in Russian, 2 vol., 195556), gives a good elementary introduction. Rafael Artzy, Geometry: An Algebraic Approach (1992), is also of interest. Hilda P. Hudson, Ruler & Compasses (1916), is a thorough treatment of constructions. The theory of congruence is covered in J.F. Rigby, Axioms for Absolute Geometry, Canadian Journal of Mathematics, 20:158181 (1968). Books on convex geometry include T. Bonnesen and W. Fenchel, Theory of Convex Bodies (1987; originally published in German, 1934), excellent for classical theory; Harold Gordon Eggleston, Convexity (1958, reprinted with corrections, 1969); Frederick A. Valentine, Convex Sets (1964, reprinted 1975); I.M. Yaglom (I.M. Iaglom) and V.G. Boltyanskii (V.G. Boltianskii, Convex Figures (1961; originally published in Russian, 1951); and Steven R. Lay, Convex Sets and Their Applications (1982, reprinted 1992). Books dealing with specialized topics in convexity include Herbert Busemann, Convex Surfaces (1958); R. Tyrrell Rockafellar, Convex Analysis (1970); and Jan van Tiel, Convex Analysis (1984). Some additional references on convexity are Edwin F. Beckenbach, Convex Functions, Bulletin of the American Mathematical Society, 54:439460 (1948); Russell V. Benson, Euclidean Geometry and Convexity (1966); Branko Grnbaum, Convex Polytopes (1967); Hugo Hadwiger and Hans Debrunner, Combinatorial Geometry in the Plane (1964; originally published in German, 1959); Edwin E. Moise, Elementary Geometry from an Advanced Standpoint, 3rd ed. (1990); C.A. Rogers, Packing and Covering (1964); Marcel Berger, Convexity, The American Mathematical Monthly, 97(8):650678 (1990), a survey of the field; and Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy, Unsolved Problems in Geometry (1991), a presentation of problems including background material, possible approaches, and references. Henry George Forder Frederick Albert Valentine The Editors of the Encyclopdia Britannica Transformation geometry Translation and reflection in the plane If, in a plane, l is a given line, P a point not on it, and F the foot of the perpendicular to l from P, and if PF is prolonged to Q so that PF FQ, then Q is called the image of P by reflection in l. If P describes a line, so does Q. If P describes a circle clockwise, then Q describes a circle anticlockwise. The transformation is called indirect because it reverses sense. Figure 14: Reflection of a point A in a line l (to yield point B) followed by The result of a reflection in a line l followed by one in m is direct, because sense is preserved. If lm the result is a translation taking any point P into a point Q, the distance of which from P is twice that between l and m. If, as in Figure 14, l and m meet in O, the result is a rotation around O through an angle twice that from l to m. Conversely, any translation or rotation can be resolved into reflections in two lines, either of which may be any line perpendicular to the direction of the translation or a line through the centre of the rotation. The other is then fixed. A displacement in a plane is a direct transformation that transforms any figure F into a congruent figure F. If F is reflected in a line, the resulting transformation is an indirect displacement. A direct displacement is always a translation or a rotation; an indirect displacement is the result of a translation and a reflection in a line parallel to the direction of the translation. Homotheties If O is a point in a plane, a transformation that leaves O fixed and changes any two other points A, B into A, B on OA, OB, respectively, so that ABAB, is called a homothety, with centre O. A may be on the ray OA or on the opposite ray. If C becomes C, then AB | BC ~ AB | BC by similar triangles and theorem B above. If two nonconcentric circles have centres A, B and unequal radii r, s, there are points I, E with , AI | IB ~ AE | EB ~ r | s. In consequence, the circles are homothetic in two ways with I as the internal and E as the external centre of homothety. These points may both lie inside the circles; but, when each circle lies outside the other, I and E are points at which pairs of common tangents meet. For example: If ABC is a triangle, and D, E, F are the midpoints of the sides, the internal centre of homothety of the circles ABC, DEF is the point of concurrence of AD, BE, CF and the external centre is the point of concurrence of the altitudes of the triangle ABC. The circle DEF is the nine-point circle of triangle ABC. The result of any sequence of homotheties and translations is a homothety or a translation.