By Julian Lowell Coolidge

Full, authoritative historical past of the thoughts for facing geometric equations covers improvement of projective geometry from old to fashionable instances, explaining the unique works, commenting at the correctness and directness of proofs, and exhibiting the relationships among arithmetic and different highbrow advancements. 1940 edition.

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Additional info for A History of Geometrical Methods

Sample text

If the equation of the locus is given in homogeneous coordinates, the process is essentially the same, for since x3 can never be zero, and since only the ratios x1/x3 and x2/x3 are significant, we can, for convenience, take x3 to be 1, in which case x1 and x2 become x and y, respectively, and the problem is reduced to the familiar one in nonhomogeneous coordinates. Now that we have introduced coordinates for lines as well as points, it is clearly possible to consider an equation in line-coordinates, say and plot it very much as we plot an equation in point-coordinates.

What is the image of the family of parallel lines y = x + k in the scene? What is the image of the circle, Γ, whose equation in the plane of the scene is x2 + (y – 2)2 = 1? It will be helpful in solving this problem to introduce two auxiliary coordinate systems in addition to the basic x, y, z system itself. One of these, an X, Y system in the xy plane, we shall need in order to describe configurations which are limited to the object plane, z = 0. The other, an X′, Z′ system in the xz plane, we shall need in order to describe configurations which are limited to the image plane, y = 0 (Fig.

6. Without using the a priori knowledge that such triangles do not exist in euclidean geometry, explain why it is impossible to find a plane perspective which will transform a given triangle into one which has two right angles. 7. Can an arbitrary segment be transformed by a plane perspective into a segment of prescribed length? 8. Can an arbitrary pentagon be transformed by a plane perspective into a regular pentagon? 9. Is it possible to find a plane perspective which will simultaneously transform two given triangles into isosceles triangles?