12 November 2023

Geometrical Figures XIX: On Circumference

"Now discourse is necessarily limited by its point of departure and its point of arrival, and since these are in mutual opposition we speak of contradiction. For the discursive reason these terms are opposed and distinct. In the realm of the reason, therefore, there is a necessary disjunction between extremes, as, for example, in the rational definition of the circle where the lines from the center to the circumference are equal and where the center cannot coincide with the circumference." (Nicholas of Cusa, "Apologia Doctae ignorantiae" ["The Defense of Learned Ignorance"], 1449)

"Many errors, of a truth, consist merely in the application of the wrong names of things. For if a man says that the lines which are drawn from the centre of the circle to the circumference are not equal, he understands by the circle, at all events for the time, something else than mathematicians understand by it." (Baruch Spinoza, "Ethics", Book I, 1677)

"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)

"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." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"The digits of pi beyond the first few decimal places are of no practical or scientific value. Four decimal places are sufficient for the design of the finest engines; ten decimal places are sufficient to obtain the circumference of the earth within a fraction of an inch if the earth were a smooth sphere" (Petr Beckmann, "A History of Pi", 1976)

"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 mathematician's circle, with its infinitely thin circumference and a radius that remains constant to infinitely many decimal places, cannot take physical form. If you draw it in sand, as Archimedes did, its boundary is too thick and its radius too variable." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"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 engineer and the mathematician have a completely different understanding of the number pi. In the eyes of an engineer, pi is simply a value of measurement between three and four, albeit fiddlier than either of these whole numbers. [...] Mathematicians know the number pi differently, more intimately. What is pi to them? It is the length of a circle’s round line (its circumference) divided by the straight length (its diameter) that splits the circle into perfect halves. It is an essential response to the question, ‘What is a circle?’ But this response – when expressed in digits – is infinite: the number has no last digit, and therefore no last-but-one digit, no antepenultimate digit, no third-from-last digit, and so on." (Daniel Tammet, "Thinking in Numbers" , 2012)

"It just so happens that π can be characterised precisely without any reference to decimals, because it is simply the ratio of any circle’s circumference to its diameter. Likewise can be characterised as the positive number which squares to 2. However, most irrational numbers can’t be characterised in this way." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

"Mathematically, circles embody change without change. A point moving around the circumference of a circle changes direction without ever changing its distance from a center. It’s a minimal form of change, a way to change and curve in the slightest way possible. And, of course, circles are symmetrical. If you rotate a circle about its center, it looks unchanged. That rotational symmetry may be why circles are so ubiquitous. Whenever some aspect of nature doesn’t care about direction, circles are bound to appear. Consider what happens when a raindrop hits a puddle: tiny ripples expand outward from the point of impact. Because they spread equally fast in all directions and because they started at a single point, the ripples have to be circles. Symmetry demands it." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

"Pi is fundamentally a child of calculus. It is defined as the unattainable limit of a never-ending process. But unlike a sequence of polygons steadfastly approaching a circle or a hapless walker stepping halfway to a wall, there is no end in sight for pi, no limit we can ever know. And yet pi exists. There it is, defined so crisply as the ratio of two lengths we can see right before us, the circumference of a circle and its diameter. That ratio defines pi, pinpoints it as clearly as can be, and yet the number itself slips through our fingers." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)

"The incommensurability of the diagonal of a square was initially a problem of measuring length but soon moved to the very theoretical level of introducing irrational numbers. Attempts to compute the length of the circumference of the circle led to the discovery of the mysterious number. Measuring the area enclosed between curves has, to a great extent, inspired the development of calculus." (Heinz-Otto Peitgen et al, "Chaos and Fractals: New Frontiers of Science" 2nd Ed., 2004)

"Imagine a person with a gift of ridicule [He might say] First that a negative quantity has no logarithm [ln(-1)]; secondly that a negative quantity has no square root [√-1]; thirdly that the first non-existent is to the second as the circumference of a circle is to the diameter [π]." (Augustus De Morgan) [attributed]

"Ten decimal places of pi are sufficient to give the circumference of the earth to a fraction of an inch, and thirty decimal places would give the circumference of the visible universe to a quantity imperceptible to the most powerful microscope." (Simon Newcomb)

"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)

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