Figurative Figures IV: On Circumference

"Nature is an infinite sphere of which the center is everywhere and the circumference nowhere." (Blaise Pascal, "Pensées", 1670)

"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)

"A system of nature proceeding from subjects of the most simple organization to such as are more perfect, or from the circumference to the centre, is called a Mathematical System." (John Lindley, "Some Account of the Spherical and Numerical System of Nature o/M. Elias Fries", ‘Philosophical magazine: a journal of theoretical, experimental and applied physics’ Vol. 68, 1826)

"Philosophical systems do not depend upon individual productions which are subject to continual variation, but upon eternal and unchangeable ideas. These always proceed from the centre to the circumference, or from the most perfect productions to those of a lower order." (John Lindley, "Some Account of the Spherical and Numerical System of Nature o/M. Elias Fries", ‘Philosophical magazine: a journal of theoretical, experimental and applied physics’ Vol. 68, 1826)

"The world can no more have two summits than a circumference can have two centres." (Pierre T de Chardin, "The Divine Milieu", 1960)

"An observer of our biological sciences today sees dark figures moving over a bridge of glass. We are faced with an ever expanding universe of light and darkness. The greater the circle of understanding becomes, the greater is the circumference of surrounding ignorance." (Erwin Chargaff, "Essays on Nucleic Acids", 1963)

"The universe has no circumference, for if it had a center and a circumference there would be some and some thing beyond the world, suppositions which are wholly lacking in truth. Since, therefore, it is impossible that the universe should be enclosed within a corporeal center and corporeal boundary, it is not within our power to understand the universe, whose center and circumference are God. And though the universe." (Nicholas of Cusa[Nicolaus Cusanus])

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)

Geometrical Figures XI: Squaring the Circle

"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." (Felix Klein, "Lectures on Mathematics", 1911)

"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." (Felix Klein, "Lectures on Mathematics", 1911)

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

[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)

Figurative Figures I: Circles

"[...] because the origin of arts and sciences is to be considered according to the present revolution of the universe, we must affirm, in conformity with the most general tradition, that geometry was first invented by the Egyptians, deriving its origin from the mensuration of their fields: since this, indeed, was necessary to them, on account of the inundation of the Nile washing away the boundaries of land belonging to each. Nor ought It to seem wonderful, that the invention of this as well as of other sciences, should receive its commencement from convenience and opportunity. Since whatever is carried in the circle of generation proceeds from the imperfect to the perfect." (Proclus Lycaeus, cca 5th century)

"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)

"Each of the parts of philosophy is a philosophical whole, a circle rounded and complete in itself. In each of these parts, however, the philosophical Idea is found in a particular specificality or medium. The single circle, because it is a real totality, bursts through the limits imposed by its special medium, and gives rise to a wider circle. The whole of philosophy in this way resembles a circle of circles. The Idea appears in each single circle, but, at the same time, the whole Idea is constituted by the system of these peculiar phases, and each is a necessary member of the organisation." (Georg W F Hegel, "Encyclopedia of the Philosophical Sciences", 1816)

"The life of a man is a self-evolving circle, which, from a ring imperceptibly small, rushes on all sides outwards to new and larger circles, and that without end. The extent to which this generation of circles, wheel without wheel, will go, depends on the force or truth of the individual soul." (Ralph W Emerson, "Circles", 1841)

"The generalizations of science sweep on in ever-widening circles, and more aspiring flights, through limitless creation." (Thomas H Huxley, [letter] 1859)

"Few will deny that even in the first scientific instruction in mathematics the most rigorous method is to be given preference over all others. Especially will every teacher prefer a consistent proof to one which is based on fallacies or proceeds in a vicious circle, indeed it will be morally impossible for the teacher to present a proof of the latter kind consciously and thus in a sense deceive his pupils. Notwithstanding these objectionable so-called proofs, so far as the foundation and the development of the system is concerned, predominate in our textbooks to the present time. Perhaps it will be answered, that rigorous proof is found too difficult for the pupil’s power of comprehension. Should this be anywhere the case, - which would only indicate some defect in the plan or treatment of the whole, - the only remedy would be to merely state the theorem in a historic way, and forego a proof with the frank confession that no proof has been found which could be comprehended by the pupil; a remedy which is ever doubtful and should only be applied in the case of extreme necessity. But this remedy is to be preferred to a proof which is no proof, and is therefore either wholly unintelligible to the pupil, or deceives him with an appearance of knowledge which opens the door to all superficiality and lack of scientific method." (Hermann G Grassmann, "Stücke aus dem Lehrbuche der Arithmetik", 1861)

"Everything in nature is a puzzle until it finds its solution in man, who solves it in some way with God, and so completes the circle of creation." (Theodore T Munger, "The Appeal to Life", 1891)

"The study of mathematics - from ordinary reckoning up to the higher processes - must be connected with knowledge of nature, and at the same time with experience, that it may enter the pupil’s circle of thought." (Johann F Herbart, "Letters and Lectures on Education", 1908)

"Human knowledge is not (or does not follow) a straight line, but a curve, which endlessly approximates a series of circles, a spiral. Any fragment, segment, section of this curve can be transformed (transformed one-sidedly) into an independent, complete, straight line [...]" (Vladimir I Lenin, "On the Question of Dialectics", 1915)

"But the star-glistered salver of infinity, The circle, blind crucible of endless space, Is sluiced by motion,-subjugated never." (Hart Crane. "The Bridge", 1930)

"Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature." (Albert Einstein, [Obituary for Emmy Noether], 1935)

"As our mental eye penetrates into smaller and smaller distances and shorter and shorter times, we find nature behaving so entirely differently from what we observe in visible and palpable bodies of our surroundings that no model shaped after our large-scale experiences can ever be ‘true’. A complete satisfactory model of this type is not only practically inaccessible, but not even thinkable. Or, to be precise, we can, of course, think of it, but however we think it, it is wrong; not perhaps quite as meaningless as a ‘triangular circle’, but more so than a ‘winged lion’." (Erwin Schrödinger, "Science and Humanism", 1952)

"The inner circle of creative mathematicians have the well-kept trade secret that in a great many cases theorems come first and axioms second." (Carl B Allendoerfer, "The Narrow Mathematician", The American Mathematical Monthly, 1962)

"As mechanics is the science of motions and forces, so thermodynamics is the science of forces and entropy. What is entropy? Heads have split for a century trying to define entropy in terms of other things. Entropy, like force, is an undefined object, and if you try to define it, you will suffer the same fate as the force definers of the seventeenth and eighteenth centuries: Either you will get something too special or you will run around in a circle." (Clifford Truesdell, "Six Lectures on Modern Natural Philosophy", 1966)

"Unless we can explain the mind in terms of things that have no thoughts or feelings of their own, we'll only have gone around in a circle." (Marvin Minsky, "The Society of Mind", 1987) 

"We move from part to whole and back again, and in that dance of comprehension, in that amazing circle of understanding, we come alive to meaning, to value, and to vision: the very circle of understanding guides our way, weaving together the pieces, healing the fractures, mending the torn and tortured fragments, lighting the way ahead - this extraordinary movement from part to whole and back again, with healing the hallmark of each and every step, and grace the tender reward." (Ken Wilber, "The Eye of Spirit: An Integral Vision for a World Gone Slightly Mad", 1997)

"Our simplistic cause-effect analyses, especially when coupled with the desire for quick fixes, usually lead to far more problems than they solve - impatience and knee-jerk reactions included. If we stop for a moment and take a good look our world and its seven levels of complex and interdependent systems, we begin to understand that multiple causes with multiple effects are the true reality, as are circles of causality-effects." (Stephen G Haines, "The Managers Pocket Guide to Systems Thinking & Learning", 1998)

"Like works of literature, mathematical ideas help expand our circle of empathy, liberating us from the tyranny of a single, parochial point of view. Numbers, properly considered, make us better people." (Daniel Tammet, "Thinking in Numbers", 2012)

"Ideas of simplicity, perfection, balance, harmony, purity, and beauty are not hard to derive from the symmetrical stability of a circle, the simplest of all geometric shapes. [...] The impression of purity and virtue might also explain why this evocative shape has been used in the iconography of major religions [...]. Another popular association with the circle is the concept of unity, wholeness, completeness, inclusion, or containment. This follows naturally from the form of a circle, a joined curve that creates two areas: interior versus exterior, inclusion versus exclusion. As such, the circle powerfully embodies ideas of boundary." (Manuel Lima, "The Book of Circle: Visualizing Spheres of Knowledge", 2017)

"The circle is a powerful symbol of generative force, associated over the ages with ideas of movement, rotation, transformation, cyclicality, and periodicity. A circle can be described as the curve drawn by a moving point revolving at a constant distance around a stationary point. This definition is central to the idea of rotation implicit in the circle and reinforced by one of the circle's inescapable manifestations, the wheel." (Manuel Lima, "The Book of Circle: Visualizing Spheres of Knowledge", 2017)

"If full knowledge about the very base of our existence could be described as a circle, the best we can do is to arrive at a polygon." (Nicholas of Cusa)

"Just as the stone thrown into the water becomes the centre and cause of various circles, [so] the sound made in the air spreads out in circles and fills the surrounding parts with an infinite number of images of itself." (Leonardo da Vinci)

"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)

Ernst W Hobson - Collected Quotes

"A great department of thought must have its own inner life, however transcendent may be the importance of its relations to the outside. No department of science, least of all one requiring so high a degree of mental concentration as Mathematics, can be developed entirely, or even mainly, with a view to applications outside its own range. The increased complexity and specialisation of all branches of knowledge makes it true in the present, however it may have been in former times, that important advances in such a department as Mathematics can be expected only from men who are interested in the subject for its own sake, and who, whilst keeping an open mind for suggestions from outside, allow their thought to range freely in those lines of advance which are indicated by the present state of their subject, untrammelled by any preoccupation as to  applications to other departments of science." (Ernst W Hobson, Nature Vol. 84, [address] 1910)

"Much of the skill of the true mathematical physicist and of the mathematical astronomer consists in the power of adapting methods and results carried out on an exact mathematical basis to obtain approximations sufficient for the purposes of physical measurements." (Ernst W Hobson, Nature Vol. 84, [address] 1910)

"Perhaps the least inadequate description of the general scope of modern Pure Mathematics - I will not call it a definition - would be to say that it deals with form, in a very general sense of the term; this would include algebraic form, functional relationship, the relations of order in any ordered set of entities such as numbers, and the analysis of the peculiarities of form of groups of operations." (Ernst W Hobson, Nature Vol. 84, [address] 1910)

"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)

"On the other side of the subject, Geometry is an abstract rational Science which deals with the relations of objects that are no longer physical objects, although these ideal objects, points, straight lines, circles, &c., are called by the same names by which we denote their physical counterparts. At the base of this rational Science there lies a set of definitions and postulations which specify the nature of the relations between the ideal objects with which the Science deals. These postulations and definitions were suggested by our actual spatial perceptions, but they contain an element of absolute exactness which is wanting in the rough data provided by our senses. The objects of abstract Geometry possess in absolute precision properties which are only approximately realized in the corresponding objects of physical Geometry." (Ernest W Hobson, "Squaring the circle", 1913)

"The number was first studied in respect of its rationality or irrationality, and it was shown to be really irrational. When the discovery was made of the fundamental distinction between algebraic and transcendental numbers, i. e. between those numbers which can be, and those numbers which cannot be, roots of an algebraical equation with rational coefficients, the question arose to which of these categories the number π belongs. It was finally established by a method which involved the use of some of the most modern of analytical investigation that the number π was transcendental. When this result was combined with the results of a critical investigation of the possibilities of a Euclidean determination, the inferences could be made that the number π, being transcendental, does not admit of a construction either by a Euclidean determination, or even by a determination in which the use of other algebraic curves besides the straight line and the circle are permitted." (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)

"The objects of abstract Geometry possess in absolute precision properties which are only approximately realized in the corresponding objects of physical Geometry." (Ernest W Hobson, "Squaring the Circle", 1913)

"We may be thinking out a chain of reasoning in abstract Geometry, but if we draw a figure, as we usually must do in order to fix our ideas and prevent our attention from wandering owing to the difficulty of keeping a long chain of syllogisms in our minds, it is excusable if we are apt to forget that we are not in reality reasoning about the objects in the figure, but about objects which ore their idealizations, and of which the objects in the figure are only an imperfect representation. Even if we only visualize, we see the images of more or less gross physical objects, in which various qualities irrelevant for our specific purpose are not entirely absent, and which are at best only approximate images of those objects about which we are reasoning." (Ernest W Hobson, "Squaring the Circle", 1913)