David Gans

Attempts to Prove the Fifth Postulate

This chapter presents the attempts to prove Euclids fifth postulate. The geometers who criticized Postulate 5 over the centuries did not question that its content was a mathematical fact, but only that it was not brief, simple, and self-evident, as postulates were supposed to be. One of the earliest known critics of Euclid was Posidonius , who found fault with Euclids definition of parallels, as well as with Euclids Postulate 5. The chapter explains that as G. Saccheri was working with different basis from that of Euclid, he reached some unfamiliar conclusions. Two parallels are not equidistant from one another. When they have a common perpendicular, they recede from one another on each side of this perpendicular. When they have no common perpendicular, they recede from each other in one direction and are asymptotic in the other direction. The last notable attempts to prove Postulate 5 were those of A. M. Legendre, the French mathematician. Throughout the 19th century non-Euclidean geometry remained a subject of great interest and study.

Single Elliptic Geometry

This chapter discusses single elliptic geometry. Single elliptic geometry resembles double elliptic geometry in that straight lines are finite and there are no parallel lines, but it differs from it in that two straight lines meet in just one point and two points always determine only one straight line. It resembles Euclidean and hyperbolic geometry. It is sometimes called elliptic geometry of the hemispherical type because of its relation to the geometry on a hemisphere. The chapter presents some basic facts of single elliptic geometry, such as:(1) each pair of straight lines meet in exactly one point, (2) through each pair of points there passes exactly one straight line, and (3) through each point there pass infinitely many straight lines, the totality of whose points constitutes the single elliptic plane. The single elliptic plane is a metric space containing at least two points, at least one simple arc of finite length joining each two points, and at least two straight lines.

Double Elliptic Geometry

This chapter discusses double elliptic geometry. In Euclidean and hyperbolic geometry, the existence of parallel lines is a consequence of the theorem that states that the exterior angle of a triangle exceeds each opposite interior angle, and this theorem is a consequence of the assumption that straight lines are infinite. Hence, a system of geometry whose straight lines are finite could not have any parallel lines at all. A system where any two straight lines meet in two points is called double elliptic geometry. It is also known as elliptic geometry of the spherical type because one is made aware of its existence by considering relations involving the entire sphere. The largest circles on a sphere are those whose radii equal the radius of the sphere. They are called great circles. The concept of a two-dimensional system of geometry that, such as Euclidean and hyperbolic geometry, employs the terms point, straight line, plane, straight line segment, distance, angle, congruent, perpendicular, and so forth, and whose points, straight lines, and straight line segments have the same properties and relations as the points, great circles, and geodesic arcs on a sphere, is the concept of double elliptic plane geometry.

Euclid's Fifth Postulate

This chapter discusses Euclids fifth postulate. It presents Euclids theory of parallels in which two straight lines are parallel if they are in the same plane and do not meet. Straight lines parallel to the same straight line are also parallel to one another. Many notable facts of Euclidean geometry besides the properties of parallel lines are consequences of Postulate 5. Among them are the Pythagorean Theorem, the formulas for the circumference and area of a circle, the fact that through any three noncollinear points there passes a circle, the existence of similar figures that are not congruent, the formula for the sum of the angles of a triangle, and the numerous consequences of that formula. The chapter presents substitutes for Postulate 5 and proofs of some of them.

VI – Triangle Relations

Publisher Summary This chapter discusses the derivation of the formulas of hyperbolic geometry, which express the numerical relations among the sides and angles of a triangle. Each of these formulas is analogous to some familiar Euclidean formula. Corresponding to the right triangle formulas a2 + b2 = c2 and sin A = a/c. The chapter discusses properties of associated right triangles. It presents the theorem that states that the sine of an acute angle in a right triangle is equal to the hyperbolic sine of the opposite side divided by the hyperbolic sine of the hypotenuse. The hyperbolic cosine of the hypotenuse of a right triangle is equal to the product of the hyperbolic cosines of the other sides. The measures of very small figures in hyperbolic geometry fit the formulas of Euclidean geometry very closely and any desired precision of it can be obtained by taking the figures sufficiently small. Euclidean geometry applies very well to the physical world of experience and is, therefore, extensively used in science, engineering, and many other fields dealing with geometrical concepts.

Parallels Without a Common Perpendicular

In hyperbolic geometry there are two kinds of parallel lines, those with a common perpendicular, and those without a common perpendicular. This chapter describes separation properties, subdivision properties, intersection properties, and miscellaneous properties of the parallel lines with a common perpendicular. A line joining two nonadjacent vertices of a quadrilateral subdivides the angles at those vertices. A line that subdivides an angle of a triangle intersects the opposite side, that is, meets it in an inner point. Two triangles are congruent if their sides are equal, respectively, or if two sides and the included angle of one triangle are equal to two sides and the included angle of the other. The chapter presents the initial theorems of hyperbolic geometry. It presents a theorem that states that the summit angles of a Saccheri quadrilateral are equal. It also presents a theorem that states that the line joining the midpoints of the base and summit of a Saccheri quadrilateral is perpendicular to each. The base and summit, therefore, lie on parallel lines having a common perpendicular.