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Extra resources for A treatise of Archimedes: Geometrical solutions derived from mechanics

Sample text

We in a definite order, and shall lie in The one simple look upon the points as being arranged no three consecutive points specify that straight line. i points in which two consecutive sides intersect. The -point and ^-side are reciprocal joining two non-consecutive vertices figures. e. diagonals) of a of of intersection non-consecutive sides the points simple -point, in the n-side are reciprocal, each to each. to 47. In metric geometry where by an //-point of a plane enclosed by ^-sides, the re-entrant is meant a portion -point, such as the THE PRINCIPLE OF RECIPROCITY OR DUALITY.

The totality of points and lines which are 1 8. said to in a plane system or field ; plane we name a and the the base of points system of plane is In the plane field there are contained, consequently, not contained the lines. only points and many ranges of straight lines points as elements, and sheaves of rays but also indefinitely ; for all the points GEOMETRY OF 14 on a lying straight and of points, all line the POSITION. of the field taken together form a range of the field passing through one lines The plane field is, therefore, justly point form a sheaf of rays.

Thus the infinitely distant 24. point lies out in both directions f upon the line, or as properly in the one direction as in the other, and the straight line appears to be closed, its extremities meeting in the infinitely distant *The assumption of a single infinitely distant point on a straight line and the definition of parallel lines as lines which intersect in a common infinitely With distant point is equivalent to the assumption of Euclid s twelfth axiom. this axiom as starting point, Euclid proves that one and only c straight can be drawn through a given point parallel to a given straight line.

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