Text. 8. Early Geometry

For a long time geometry was intimately tied to physical space, actually beginning as a gradual accumulation of subconscious notions about physical space and about forms, content, and spatial relations of specific objects in that space. We call this very early geometry "subconscious geometry". Later, human intelligence evolved to the point where it became possible to consolidate some of the early geometrical notions into a collection of somewhat general laws or rules. We call this laboratory phase in the development of geometry "scientific geometry". About 600 B.C. the Greeks began to inject deduction into geometry giving rise to what we call “demonstrative geometry”.

In time demonstrative geometry becomes a material-axiomatic study of idealized physical space and of the shapes, sizes, and relations of idealized physical objects in that space. The Greeks had only one space and omgeometry; these were absolute concepts. The space was not thought of as collection of points but rather as a realm or locus, in which objects could be freely moved about and compared with one another. From this point of view the basic relation in geometry was that of congruence or superposability.

With the elaboration of analytic geometry in the first half of the seventeenth century, space came to be regarded as a collection of points; and wit: the invention, about two hundred years later, of the classical non- Euclidean geometries, mathematicians accepted the situation that there is more than one conceivable space and hence more than one geometry. But space was still regarded as a locus in which figures could be compared with one another. The central idea became of a group of congruent transformations of space into itself and geometry came to be regarded as the study of those properties of configurations of points which remain unchanged when the enclosing space is subjected to these transformations, and geometry is defined as the invariant theory of a transformation group. Geometry came to be rather far removed from its former intimate connection with physical space, and it became a relatively simple matter to invent new and even bizzare geometries

Vocabulary

1. Answer the following questions:

1) How was space regarded in the 17^th century?

2) What removed geometry far from its former intimate connection with physical space?

3) How many dimensions did the Greeks have?

4) Why was early geometry called “subconscious geometry”?

5) How was the space thought of by the Greeks?

6) What give rise to “demonstrative geometry”?

2. Give Russian equivalents to the English ones:

From this point of view, geometry came to be regarded as, to be far removed from, to move freely, to give rise to, human intelligence, general rules, to inject deduction into, to evolve to the point, to be intimately tied, early geometrical notions, study of shapes and sizes, the space was thought of, to consolidate some notions, to be subjected to transformations.