Two important geometries alternative to

Euclidean geometry are elliptic geometry and hyperbolic geometry.

For instance, the number of assertions in the

Euclidean geometry is infinite.

In [1] it is noted that Tarski's system of foundations of geometry has a number of distinctive features, in which it differs from most, if not all, systems of foundation of

Euclidean geometry that are known from the literature.

Note, further, that once one has grasped the meaning of ratio and proportion, the insight readily can be repeated for a range of problems pertaining, for example, to classical

Euclidean geometry, coordinate equations of lines, or even production ratios in an economy.

In fact, they all belong to the

Euclidean geometry of the sphere--a body of truths that was known and used by the Greek astronomers who believed that the stars were embedded in a great sphere rotating around the earth.

To introduce your students to elastic geometry, and to help them realize why this topic is different from

Euclidean geometry, have them draw a circle, first on a piece of paper, then on a piece of latex.

This was a matter of fact: the idea of the transcendental intuition of space is irreconcilable with the simultaneous truth and presence of a

Euclidean geometry and a non-

Euclidean geometry with the same rights to existence, the same rights to citizenship in the realm of the episteme.

Greenberg outlines the essence of

Euclidean geometry and shows how the hyperbolic and elliptic, non-Euclidean, geometries have developed in modem times.

Some of Hoffe's most suggestive analyses occur in this part of the book when he argues that Kant's theory of space is not tied to

Euclidean geometry (p.

Egypt, the Mideast, and China all had important mathematical discoveries early in history; only the Greeks developed the rhetoric, a logical proof by axiomatic method as we see in

Euclidean geometry.

Part I, "Introduction," provides a motivation for the use of fractal geometry, as an alternative to

Euclidean geometry, in the analysis of complex natural patterns.

The elements Marcaccio cannibalizes--ground, brushstroke, color--remained passive in the post-Modern recycling of painting: "Halley makes an echo of the structure of

Euclidean geometry, a statement about institutionalized structures of power.