"Well I now apply the straight rod - so - thus squaring the circles: and there you are. In the center you have your market place: straight streets leading into it, from here, from here, from here. Very much the same principle, really, as the rays of a star: the star itself is circular, but sends out straight rays in every direction." (Aristophanes, "The Birds", cca. 414 BC)
"With the straight ruler I set to work. To make the circle work four-cornered." (Aristophanes, "The Birds", cca. 414 BC)
"The circle, is larger than every inscribed rectilinear figure, but is less than every circumscribed one. [...] But also the rectilinear figure drawn between the circumscribed and inscribed figure is smaller than the circumscribed and larger than the inscribed figure. Things larger and smaller than the same are equal to one another. Therefore, the circle is equal to the rectilinear figure drawn between the inscribed figure and the circumscribed figure. But we can construct a square equal to every given rectilinear figure. Therefore it is possible to produce a square equal to the circle." (Alexander of Aphrodisias, cca. 3rd century)
"The seventh mode has reference to distances, positions, places and the occupants of the places. In this mode things which are thought to be large appear small, square things round; flat things appear to have projections, straight things to be curved." (Diogenes Laertius, "Lives of the Philosophers", 3rd century AD)
"Mad Mathesis alone was unconfined,
Too mad for mere material claims to bind,
Now to pure space lifts her ecstatic stare,
Now, running round the circle, finds it square." (Alexander Pope, "Dunciad", 1743)
"This measure will and must prove a great benefit to mankind, when understood, as it is the basis and foundation of mathematical operations, for, without a perfect quadrature of the circle, measures, weighs, etc, must still remain hidden and unrevealed facts, which are and will be of great importance to rising generations. The improvements that will arise from this measure fifty years hence I cannot paint in imagination. (John Davis, "The Measure of the Circle", 1854)
"The gambling reasoner is incorrigible; if he would but take to the squaring of the circle, what a load of misery would be saved." (Augustus De Morgan, "A Budget of Paradoxes", 1872)
"The circumference of any circle being given, if that circumference be brought into the form of a square, the area of that square is equal to the area of another circle, the circumscribed square of which is equal to the area of the circle whose circumference is first given." (John A Parker, "The Quadrature of the Circle", 1874)
"Infinity is the land of mathematical hocus pocus. There Zero the magician is king. When Zero divides any number he changes it without regard to its magnitude into the infinitely small [great?], and inversely, when divided by any number he begets the infinitely great [small?]. In this domain the circumference of the circle becomes a straight line, and then the circle can be squared. Here all ranks are abolished, for Zero reduces everything to the same level one way or another. Happy is the kingdom where Zero rules!" (Paul Carus, "The Nature of Logical and Mathematical Thought"; Monist Vol 20, 1910)
"The proof of the transcendency of π will hardly diminish the number of circle-squarers, however; for this class of people has always shown an absolute distrust of mathematicians and a contempt for mathematics that cannot be overcome by any amount of demonstration."
"The proof that π is a transcendental number will forever mark an epoch in mathematical science. It gives the final answer to the problem of squaring the circle and settles this vexed question once for all. This problem requires to derive the number π by a finite number of elementary geometrical processes, i.e. with the use of the ruler and compasses alone. As a straight line and a circle, or two circles, have only two intersections, these processes, or any finite combination of them, can be expressed algebraically in a comparatively simple form, so that a solution of the problem of squaring the circle would mean that π can be expressed as the root of an algebraic equation of a comparatively simple kind, viz. one that is solvable by square roots."
"Every Scientific Society still receives from time to time communications from the circle squarer and the trisector of angles, who often make amusing attempts to disguise the real character of their essays. The solutions propounded by such persons usually involve some misunderstanding as to the nature of the conditions under which the problems are to be solved, and ignore the difference between an approximate construction and the solution of the ideal problem." (Ernest W Hobson, "Squaring the circle", 1913)
"The popularity of the problem among non-Mathematicians may seem to require some explanation. No doubt, the fact of its comparative obviousness explains in part at least its popularity; unlike many Mathematical problems, its nature can in some sense be understood by anyone; although, as we shall presently see, the very terms in which it is usually stated tend to suggest an imperfect apprehension of its precise import. The accumulated celebrity which the problem attained, as one of proverbial difficulty, makes it an irresistible attraction to men with a certain kind of mentality. An exaggerated notion of the gain which would accrue to mankind by a solution of the problem has at various times been a factor in stimulating the efforts of men with more zeal than knowledge. The man of mystical tendencies has been attracted to the problem by a vague idea that its solution would, in some dimly discerned manner, prove a key to a knowledge of the inner connections of things far beyond those with which the problem is immediately connected." (Ernest W Hobson, "Squaring the Circle", 1913)
"The solutions propounded by the circle squarer exhibit every grade of skill, varying from the most futile attempts, in which the writers shew an utter lack of power to reason correctly, up to approximate solutions the construction of which required much ingenuity on the part of their inventor. In some cases it requires an effort of sustained attention to find out the precise point ill the demonstration at which the error occurs, or in which an approximate determination is made to do duty for a theoretically exact one." (Ernest W Hobson, "Squaring the circle", 1913)
"To square a circle means to find a square whose area is equal to the area of a given circle. In its first form this problem asked for a rectangle whose dimensions have the same ratio as that of the circumference of a circle to its radius. The proof of the impossibility of solving this by use of ruler and compasses alone followed immediately from the proof, in very recent times, that π cannot be the root of a polynomial equation with rational coefficients."
[Mathematician:] "A scientist who can figure out anything except such simple things as squaring the circle and trisecting an angle." (Evan Esar, "Esar's Comic Dictionary", 1943)
"The squaring of the circle is a stage on the way to the unconscious, a point of transition leading to a goal lying as yet unformulated beyond it. It is one of those paths to the centre." (Carl G Jung, "Psychology and Alchemy", 1944)
"There are a number of diagrams in the literature of Sacred Geometry all related to the single idea known as the 'Squaring of the Circle'. This is a practice which seeks, with only the usual compass and straight-edge, to construct a square which is virtually equal in perimeter to the circumference of a given circle, or which is virtually equal in area to the area of a given circle. Because the circle is an incommensurable figure based on π, it is impossible to draw a square more than approximately equal to it." (Robert Lawlor, "Sacred Geometry", 1982)
"The Squaring of the Circle is of great importance to the geometer-cosmologist because for him the circle represents pure, unmanifest spirit-space, while the square represents the manifest and comprehensible represents world. When a near-equality is drawn between the circle and square, the infinite is able to express its dimensions or qualities through the finite." (Robert Lawlor, "Sacred Geometry", 1982)
"In the case of circle squaring, since the problem requires pinpointing the ratio between a circle’s diameter and circumference, the irrational number the investigator bumps into is pi (π). Perhaps because of its extreme (in fact, total) difficulty - similar to the alchemist’s hope of turning lead into gold - circle squaring offered its pursuers the dream of international fame in the discovery of an unknown quantity seemingly woven into the fabric of the universe." (Daniel J Cohen, "Equations from God: Pure Mathematics and Victorian Faith", 2007)
"Is it possible to construct a square, using only a compass and a straightedge, that is exactly equal in area to the area of a given circle? If π could be expressed as a rational fraction or as the root of a first- or second-degree equation, then it would be possible, with compass and straightedge, to construct a straight line exactly equal to the circumference of a circle. The squaring of the circle would quickly follow. We have only to construct a rectangle with one side equal to the circle’s radius and the other equal to half the circumference. This rectangle has an area equal to that of the circle, and there are simple procedures for converting the rectangle to a square of the same area. Conversely, if the circle could be squared, a means would exist for constructing a line segment exactly equal to π. However, there are ironclad proofs that π is transcendental and that no straight line of transcendental length can be constructed with compass and straightedge. " (Martin Gartner, "Sphere Packing, Lewis Carroll, and Reversi", 2009)
"The attempt to apply rational arithmetic to a problem in geometry resulted in the first crisis in the history of mathematics. The two relatively simple problems - the determination of the diagonal of a square and that of the circumference of a circle - revealed the existence of new mathematical beings for which no place could be found within the rational domain." (Tobias Dantzig)
No comments:
Post a Comment