"There are many useful connections between these two disciplines [geometry and algebra]. Many applications of algebra to geometry and of geometry to algebra were known in antiquity; nearer to our time there appeared the important subject of analytical geometry, which led to algebraic geometry, a vast and rapidly developing science, concerned equally with algebra and geometry. Algebraic methods are now used in projective geometry, so that it is uncertain whether projective geometry should be called a branch of geometry or algebra. In the same way the study of complex numbers, which arises primarily within the bounds of algebra, proved to be very closely connected with geometry; this can be seen if only from the fact that geometers, perhaps, made a greater contribution to the development of the theory than algebraists." (Isaak M Yaglom, "Complex Numbers in Geometry", 1968)
"In the geometry of similarities the only geometric properties of a figure are angles and ratios of distances. So in that geometry the only possible construction problems involve certain angles between lines of the figure and ratios of distances between its points." (Isaak Yaglom, "Geometric Transformations", 1973)
"A geometry is a discipline concerned with those properties ojjgures which do not change under the transformations of a group of transformations. This definition emphasizes that there are many geometries, not just one, and that to obtain a geometry we need only select a group of transformations." (Isaak Yaglom, "Geometric Transformations", 1973)
"A transformation of the plane which carries lines passing through some definite part of the plane into lines is called a projective transformation or projectivity.t Every affine transformation is a projective transformation, but the converse is not true; for example, a central projection of a plane onto itself is a projective transformation but is not, in general, affine." (Isaak Yaglom, "Geometric Transformations", 1973)
"An instance of such a 'non-Euclidean' geometry is projective geometry, concerned with those properties of figures which do not change under projective transformations. Projective geometry is not merely not Euclidean geometry; it is 'very much non-Euclidean'." (Isaak Yaglom, "Geometric Transformations", 1973)
"Elementary plane geometry concerns itself largely with figures made up of lines and circles. It can be shown that similarities can be defined as transformations of the plane which carry lines into lines and circles into circles. The transformations of the plane that preserve lines (i.e., carry lines into lines) without necessarily preserving circles are known as afine transformations or afinilies and form a group1 which is the basis of afine geometry." (Isaak Yaglom, "Geometric Transformations", 1973)
"Historically, hyperbolic geometry was first developed on an axiomatic basis. It arose as a result of efforts to prove the axiom of parallels from the other axioms. Doubt persisted for a long time as to whether this axiom could be deduced from the remaining axioms of Euclidean geometry. In their attempts to prove this axiom, mathematicians used the method of 'proof by contradiction' i.e., they assumed that the axiom of parallels was false and tried, on the basis of this assumption, to obtain a contradiction. All of these attempts were fruitless. True, the theorems obtained by negating the axiom of parallels appeared strange, but they did not contradict one another. The issue was resolved when C. F. Gauss, N. I. Lobachevski and J. Bolyai first stated explicitly that by negating the axiom of parallels one arrives at a new geometry, just as consistent as the usual (Euclidean) geometry." (Isaak Yaglom, "Geometric Transformations", 1973)
"Hyperbolic geometry is exceptional among non-Euclidean geometries because it satisfies all the axioms of Euclidean geometry except for the axiom of parallels. Other non-Euclidean geometries differ more radically from Euclidean geometry; in some of them a line segment cannot be produced indefinitely in both directions, and in others, two points cannot always be joined by a line." (Isaak Yaglom, "Geometric Transformations", 1973)
"Introduction of the line at infinity and of points at infinity enables us to include in a single proposition a number of special propositions, all provable in a similar way. This is because, insofar as central projections are concerned, the fictitious points at infinity are on an equal footing with actual points; points of one type can be carried into points of the other type." (Isaak Yaglom, "Geometric Transformations", 1973)
"We shall now compare the non-Euclidean geometry of Lobachevski-Bolyai with the geometry of Euclid studied in high school. One is immediately struck by how much the two geometries have in common. In both geometries two points determine a unique line, and two lines can have at most one point in common (this follows from the fact that lines of hyperbolic geometry are segments of lines in the plane). Further, in both geometries it is possible to carry a point and a ray issuing from it, by a motion, into any other point and a preassigned ray issuing from the latter point. The hyperbolic length of a segment and magnitude of an angle share many properties with their Euclidean counterparts; for example, in both geometries the length of the sum of two segments is the sum of their lengths, and the measure of the sum of two angles is equal to the sum of their measures." (Isaak Yaglom, "Geometric Transformations", 1973)
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