Two important geometries alternative to Euclidean geometry
are elliptic geometry and hyperbolic geometry.
For instance, the number of assertions in the Euclidean geometry
In  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
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.