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Ruler-and-compass constructions : Ruler and compassA number of ancient problems in geometry involve the construction of lengths or angles using only an idealised ruler and compass. The ruler is indeed a straightedge, and may not be marked; the compass may only be set to already constructed distances, and used to describe circular arcs.Some famous ruler-and-compass problems have been proved impossible, in several cases by the results of Galois theory. In spite of these impossibility proofs, some mathematical amateurs persist in trying to solve these problems. Many of them fail to understand that many of these problems are trivially soluble provided that other geometric transformations are allowed: for example, squaring the circle is possible using geometric constructions, but not possible using ruler and compasses alone. Mathematician Underwood Dudley has made a sideline of collecting false ruler-and-compass proofs, as well as other work by mathematical cranks, and has collected them into several books.
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The most famous of these problems, "squaring the circle", involves constructing a square with the same area as a given circle using only ruler and compasses.
Squaring the circle has been proved impossible, as it involves generating a transcendental ratio, namely 1:√π. Only algebraic ratios can be constructed with ruler and compasses alone. The phrase "squaring the circle" is often used to mean "doing the impossible" for this reason.
Without the constraint of requiring solution by ruler and compasses alone, the problem is easily soluble by a wide variety of geometric and algebraic means, and has been solved many times in antiquity.
Doubling the cube: using only ruler and compasses, construct the side of a cube that has twice the volume of a cube with a given side. This is impossible because the cube root of 2, though algebraic, cannot be computed from integers by addition, subtraction, multiplication, division, and taking square roots.
Angle trisection: using only ruler and compasses, construct an angle that is one-third of a given arbitrary angle. This requires taking the cube root of an arbitrary complex number with absolute value 1 and is likewise impossible.
Some regular polygons are easy to construct with ruler and compasses; others are not. This led to the question being posed: is it possible to construct all regular n-gons with ruler and compasses?
Carl Friedrich Gauss proved in 1796 that a regular n-gon can be constructed with ruler and compasses if and only if n is a power of 2 or the product of a power of 2 and distinct Fermat primes.
He was so pleased by this result that he requested that a regular 17-gon be inscribed on his tombstone.
It is possible, as shown by Georg Mohr, to construct anything with just a compass that can be constructed with ruler and compass. It is impossible to take a square root with just a ruler, so some things cannot be constructed with a ruler that can be constructed with a compass; but given a circle and its center, they can be constructed.
Simon Plouffe[?] has written a paper showing how ruler and compasses can be used as a simple computer with unexpected power to compute binary digits of certain numbers.
See also: Gauss-Wantzel theorem[?], Mohr-Mascheroni theorem, Poncelet-Steiner theorem[?], Squaring the circle
all. The camp of King Ferdinand was situated at Santa Fe, a few miles to
determined to get a final answer one way or the other to his question.
usual polite tactics. They appointed a junta.html">junta, which was presided over by
more the weary business was gone through, but Columbus must have had some
correspondence. It was no scruple of conscience that held him back, we
attainment of a great object; he would not have hesitated to adopt any
probable that either he was not quite sure of his ground and his courage
the members of the junta, that a favourable decision would at last be
the Fathers of the Church, and disposed of his proposals simply on the
heads; and still no mother-wit or gleam of light on this obscurity of
decision to the King and Queen. The monarchs, true to their somewhat
Columbus that at present they were too much occupied with the war to
campaign were a thing of the past, they might again turn their attention
promises had been made to him, and hope had been held out to him too
he would try France; at least he would be no worse off there. But he had
growing boy nearly eleven years old, had been staying with Beatriz at
his aunt's at Huelva before he went away. He set out with a heavy heart,
CHAPTER X
OUR LADY OF LA RABIDA
It is a long road from Santa Fe to Huelva, a long journey to make on
sudden weariness and the asking of innumerable difficult questions, would
him farther away from the glittering scene where his hopes had once been
nearer that unknown destiny as to which he was in great darkness of mind,
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