The difficulty obviously lies with the last two problems. To find a point of intersection of two straight lines.To find the points of intersection of a straight line and a circle.To find the point of intersection of two circles.To draw a circle with the given center and radius.With regard to the Mascheroni's result, instead of checking every single construction in the plane weĪgree that such constructions can be accomplished with a sequence of the four basic ones: Using only elementary Projective Geometry itĬan be shown that the center of the circle is indispensable. In his book Geometrical Constructions Using a Straight LineĪnd a Fixed Circle published in 1833, Steiner was able to prove that given a fixed circle and its center, all the constructions in the plane can be carried out by the straightedge alone. Inspired by Mascheroni's result, Jacob Steiner (1796-1863) tried to prove a similar result for a For this reason, constructions with compass only are commonly referred to as the Mohr-Mascheroni constructions. To his great surprise Hjelmslev found a complete treatment of the Mascheroni result in the first part of the book. Mohr titled Euclides Danicus (The Danish Euclid) and published in 1672 in Amsterdam. Interestingly, in 1928 the Danish mathematician Hjelmslev discovered in a bookshop in Copenhagen a bookīy G. The assertion that every ruler-and-compass construction could be accomplished with a compass is due to Lorenzo Mascheroni (1750-1800) and appeared in his 1797 tractate The Geometry of Compasses. Geometry of Compass only deals with constructions in the Euclidean plane, and its basic question could be formulated as, What ruler-and-compass constructions could be accomplished with the compass alone? The question of geometric construction with the compass alone is not concerned with such kinds of geometries. On the sphere, the role of straight lines is played by the great circles.
g., in finite geometries that only contain a finite number of points and lines, a line is just a (finite) collection of points. There are geometries in which the ruler is never used to start with. Thus in theįollowing constructing a straight line means finding two points that belong to that line. Starting with two points, other points can be constructed with compass alone. In what is known as the Geometry of Compass, a straight line is defined by any pair However, you can do everything reasonable. For example, you can't draw straight lines using a compass. Geometric Construction with the Compass AloneĮverything you can do with a ruler and a compass you can do with the compass alone.
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