Chapter1
    Euclid’s Elements, Book I
    (constructions)
                   2                         Euclid’s Elements, Book I (constructions)
                   1.1   Theuseofrulerandcompass
                   Euclid’s Elements can be read as a book on how to construct certain geometric
                   figures efficiently and accurately using ruler and compass, and ascertaining the
                   validity. The first three postulates before Book I are on the basic use of the ruler
                   and the compass.
                   Postulate 1. To draw a straight line from any point to any point.
                      With a ruler (straightedge) one connects two given points A and B to form the
                   line (segment) AB, and there is only one such line. This uniqueness is assumed, for
                   example, in the proof of I.4.
                   Postulate 2. To produce a finite straight line continuously in a straight line.
                      Given two points A and B, with the use of a ruler one can construct a point C
                   so that the line (segment) AC contains the point B.
                      Thefirsttwopostulatescanbecombinedintoasingleone: throughtwodistinct
                   points there is a unique straight line.
                  1.1 The use of ruler and compass                                          3
                  Postulate 3. To describe a circle with any center and distance.
                     This distance is given by a finite line (segment) from the center A to another
                  point B. With the use of a collapsible compass, one constructs a circle with given
                  center A to pass through B. We denote this circle by C(A,B). Euclid I.2 shows
                  how to construct a circle with a given center and radius equal to a given line (seg-
                  ment).
            4               Euclid’s Elements, Book I (constructions)
            Definition (I.20). Of trilateral figures,
            an equilateral triangle is that which has its three sides equal,
            an isosceles triangle that which has two of its sides alone equal, and
            a scalene triangle that which has its three sides unequal.
            Remark. Euclid seems to take isosceles and scalene in the exclusive sense. But it
            is more convenient to take these in the inclusive sense. An isosceles triangle is one
            with two equal sides, so that an equilateral triangle is also isosceles.