"[...] 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)
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