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II. TABLE of Correspondence of the Elements of EUCLID with these Books

of Geometry.

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ELEMENTS

OF

GEOMETRY,

GEOMETRY is that branch of natural science which treats of figured space.

The science

Our knowledge respecting external objects is grounded entirely on the information received through the medium of the senses. of physics considers bodies as they actually exist, invested at once with all their various qualities, and endued with their peculiar affections. Its researches are hence directed by that refined species of observation which is termed Experiment. Geometry takes a more limited view, and, selecting only the generic property of magnitude, it can, from the extreme simplicity of its basis, safely pursue the most lengthened train of investigation, and arrive with perfect certainty at the remotest conclusion. It contemplates merely the forms which

A

bodies present, and the spaces which they occupy. Geometry is thus likewise founded on external observation; but such observation is so familiar and obvious, that the primary notions which it furnishes might seem intuitive, and have often been regarded as innate. That science is, therefore, supereminently distinguished by the luminous evidence which constantly attends every step of its march.

PRINCIPLES.

In contemplating an external object, we can, by successive acts of abstraction, reduce the complex idea which arises in the mind into others that are progressively simpler. Body, divested of its essential characters, présents the mere idea of surface; a surface, considered apart from its peculiar qualities, only exhibits linear boundaries; and a line, abstracting its continuity, leaves nothing in the imagination but the points which form its extremities. solid is bounded by surfaces; a surface is circumscribed by lines; and a line is terminated by points. A point marks position; a line measures distance; and a surface represents extension. A line has only length; a surface has both length and breadth; and a solid combines all the three dimensions of length, breadth, and thickness.

A

The uniform description of a line which through its whole extent stretches in the same direction

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