A Brief History of Mathematics;
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274 HISTORY OF MATHEMATICS.
ometry has a common limiting case in the ordinary- or Euclidean geometry. These three geometries have this in common that they hold for surfaces of constant curvature. According as the constant value of the \:urvature is positive, zero, or negative, we have to do with spherical, Euclidean, or pseudo-spherical geom- etry.
A new presentation of the same theory is due to F. Klein. After projective geometry had shown that in projection or linear transformation all descri...ptive properties and also some metric relations of the fig- ures remain unaltered, the endeavor was made to find for the metric properties an expression which should remain invariant after a linear transformation. After a preparatory work of Laguerre which made the ''no- tion of the angle projective," Cayley, in 1859, found the general solution of this problem by considering ''every metric property of a plane figure as contained in a projective relation between it and a fixed conic." Starting from the Cayley theory, on the basis of the consideration of measurements in space, Klein suc- ceeded in showing that from the projective geometry with special determination of measurements in the plane there could be derived an elliptic, parabolic, or hyperbolic geometry,* the same fundamentally as the spherical, Euclidean, or pseudo-spherical geom- etry respectively.
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You can find similar books in the "Read Also" column, or choose other free books by Smith, David Eugene, 1860-1944 to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
274 HISTORY OF MATHEMATICS.
ometry has a common limiting case in the ordinary- or Euclidean geometry. These three geometries have this in common that they hold for surfaces of constant curvature. According as the constant value of the \:urvature is positive, zero, or negative, we have to do with spherical, Euclidean, or pseudo-spherical geom- etry.
A new presentation of the same theory is due to F. Klein. After projective geometry had shown that in projection or linear transformation all descri...ptive properties and also some metric relations of the fig- ures remain unaltered, the endeavor was made to find for the metric properties an expression which should remain invariant after a linear transformation. After a preparatory work of Laguerre which made the ''no- tion of the angle projective," Cayley, in 1859, found the general solution of this problem by considering ''every metric property of a plane figure as contained in a projective relation between it and a fixed conic." Starting from the Cayley theory, on the basis of the consideration of measurements in space, Klein suc- ceeded in showing that from the projective geometry with special determination of measurements in the plane there could be derived an elliptic, parabolic, or hyperbolic geometry,* the same fundamentally as the spherical, Euclidean, or pseudo-spherical geom- etry respectively.
What to read after A Brief History of Mathematics;?
You can find similar books in the "Read Also" column, or choose other free books by Smith, David Eugene, 1860-1944 to read onlineMoreLess
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