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DEFINITIONS—POINTS.

formed of this by making a mould round the body andwithdrawing it, when the mere sight of the mould gives usan exact notion of the space the body occupied. An emptybox encloses a portion of space, and the figure of this por-tion is precisely the same as that of the interior of the box.

Consequently, all the geometrical properties of the di-mensions of a body also belong to the space it fills. Thegeometrical properties of any surface, and those of thespace occupied at any given moment by it, are the same.*

This is the reason why a purely theoretical geometriciannever considers any particular body nor its individual sur-face, in order to ascertain the relations belonging to thedimensions either of the body or of the surface. He ima-gines, in space itself, the form and the surface of a body,which are sufficient for him. At first this species of ab-straction presents some difficulties, but it exercises themind and strengthens the imagination; in the result, itgives great powers of conception both in pure geometry,and in geometry applied to the arts. It is of much con-

* The observations in the text would lead us, apparently, to aclearer comprehension of geometrical definitions than is usually ob-tained. As the geometer always considers figured space, his doc-trines, even when most abstract, do not relate, as is sometimes sup-posed, to imaginary points and lines, but to the portions of space in-cluded within them. A surface is the extended limit of a solid, aline is the boundary of a surface, and a point is the termination of aline. Space itself has neither boundaries, limits, nor terminations ;all bodies have: and it is the relations as to form or position ofwhat is included in these bounds, or constitutes these limits andterminations, which the geometer considers. If vre, place togethertwo pieces of polished marble, or well-planed wood, closely adjustedto one another, so as to bring into contact every point of their sur-faces, the almost invisible line of separation between them will giveus an idea of a geometrical line, and the termination of that line hasbeen denominated by geometricians, a point. Now it is not thispoint, nor the line itself, nor the surface as a surface, which is everconsidered in geometry; but the properties or relations, as to formof the marble or w ood (and they may be taken in this respect as the

representatives of all similarly formed bodies),_of which the point,

the line, and the surface are the respective boundaries.

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