A New Look at Geometry (Dover Books on Mathematics) by Irving Adler

By Irving Adler

This richly certain evaluate surveys the advance and evolution of geometrical rules and ideas from precedent days to the current. as well as the connection among actual and mathematical areas, it examines the interactions of geometry, algebra, and calculus. The textual content proves many major theorems and employs numerous very important concepts. Chapters on non-Euclidean geometry and projective geometry shape short, self-contained treatments.
More than a hundred routines with solutions and 2 hundred diagrams remove darkness from the textual content. academics, scholars (particularly these majoring in arithmetic education), and mathematically minded readers will take pleasure in this striking exploration of the function of geometry within the improvement of Western clinical thought.
Introduction to the Dover variation through Peter Ruane.

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Fire, he taught, is composed of units, each of which is a regular tetrahedron; air is composed of octahedra; water is composed of icosahedra; and earth is composed of cubes. The heavenly sphere is represented in the Pythagorean doctrine by a regular dodecahedron inscribed in it. The dodecahedron, discovered by Pythagoras, and used as a symbol of the heavenly sphere, was the prime secret of the Order of the Pythagoreans. Hippasus, a member of the Order, violated his oath of membership by revealing this secret to outsiders.

The corresponding statements that express the information given in the second and the third lines are related to each other in the same way, that is, each is the dual of the other. The corresponding statement that expresses the information given in the first line is its own dual. What we have observed here is a foretaste of a more general symmetry known as duality which we shall discuss in Chapter 10. The duality revealed in the table is not accidental. It corresponds to a significant geometric relationship: The centers of the faces of a regular polyhedron are the vertices of another regular polyhedron inscribed in it.

To associate a point of the plane with every complex number a + bi, we use this procedure: At the point a on the axis of real numbers draw the line that is perpendicular to the axis of real numbers. Associate a real number scale with the points on this vertical line in such a way that the unit of length on it is the same as on the axis of real numbers, and the positive half of the scale is above the axis of real numbers. Locate on the vertical line the point whose scale number on this line is b.

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