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Of Ratios and Proportions

Of Similar Rectilinear Figures

Of the Straight Line and Circle

Of Polygons inscribed in a Circle and circumscribed about a Circle

SECTION II. OF THE MEASURE AND COMPARISON OF

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ELEMENTS OF GEOMETRY.

GENERAL NOTIONS OF EXTENSION.

1. GEOMETRY is that science which teaches us to investigate the magnitudes and forms of extended things, or extended space, and the relations of their parts.

2. A body, or the space which it occupies, is extended in three directions; it has length, breadth, and thickness. or depth. If it were destitute of either of these dimensions, it would cease to be.

3. The space which a body occupies, is separated from other space by what we call the surface or the outside of the body. A surface is extended in two directions; it has length and breadth, but is destitute of thickness, and therefore makes no part of the body itself. A body then, and even definite space, is bounded by surface.

4. If the body have several faces, like a square block of wood, for instance, these faces may be considered as so many distinct surfaces, each of which is bounded by the edges formed by the meeting of this face with the other faces of the body. These limits are no part of the surface; they have neither breadth nor thickness; they have only length, and are called lines. The limit of a surface, therefore, is a line.

5. The line itself is limited by a point; which has no extension. A point may also be taken in a line, a surface, a body, or in extended space; it, however, makes no part of either of these magnitudes. It has position, but no extent. It may, by moving in space, be considered as generating a line.

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