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Circle Quotes (53 quotes)

Every teacher certainly should know something of non-euclidean geometry. Thus, it forms one of the few parts of mathematics which, at least in scattered catch-words, is talked about in wide circles, so that any teacher may be asked about it at any moment. ... Imagine a teacher of physics who is unable to say anything about Röntgen rays, or about radium. A teacher of mathematics who could give no answer to questions about non-euclidean geometry would not make a better impression.
On the other hand, I should like to advise emphatically against bringing non-euclidean into regular school instruction (i.e., beyond occasional suggestions, upon inquiry by interested pupils), as enthusiasts are always recommending. Let us be satisfied if the preceding advice is followed and if the pupils learn to really understand euclidean geometry. After all, it is in order for the teacher to know a little more than the average pupil.
In George Edward Martin, The Foundations of Geometry and the Non-Euclidean Plane (1982), 72.
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Toutes les fois que dans une équation finale on trouve deux quantités inconnues, on a un lieu, l'extrémité de l'une d’elles décrivant une ligne droite ou courbe. La ligne droite est simple et unique dans son genre; les espèces des courbes sont en nombre indéfini, cercle, parabole, hyperbole, ellipse, etc.
Whenever two unknown magnitudes appear in a final equation, we have a locus, the extremity of one of the unknown magnitudes describing a straight line or a curve. The straight line is simple and unique; the classes of curves are indefinitely many,—circle, parabola, hyperbola, ellipse, etc.
Introduction aux Lieux Plans et Solides (1679) collected in OEuvres de Fermat (1896), Vol. 3, 85. Introduction to Plane and Solid Loci, as translated by Joseph Seidlin in David E. Smith(ed.)A Source Book in Mathematics (1959), 389. Alternate translation using Google Translate: “Whenever in a final equation there are two unknown quantities, there is a locus, the end of one of them describing a straight line or curve. The line is simple and unique in its kind, species curves are indefinite in number,—circle, parabola, hyperbola, ellipse, etc.”
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A grove of giant redwoods or sequoias should be kept just as we keep a great or beautiful cathedral. The extermination of the passenger pigeon meant that mankind was just so much poorer; exactly as in the case of the destruction of the cathedral at Rheims. And to lose the chance to see frigate-birds soaring in circles above the storm, or a file of pelicans winging their way homeward across the crimson afterglow of the sunset, or a myriad terns flashing in the bright light of midday as they hover in a shifting maze above the beach—why, the loss is like the loss of a gallery of the masterpieces of the artists of old time.
In A Book-Lover's Holidays in the Open (1916), 316-317.
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A human being is part of the whole, called by us “Universe”; a part limited in time and space. He experiences himself, his thoughts and feelings as something separated from the rest—a kind of optical delusion of his consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest us. Our task must be to free ourselves from this prison by widening our circle of compassion to embrace all living creatures and the whole of nature in its beauty. Nobody is able to achieve this completely but the striving for such achievement is, in itself, a part of the liberation and a foundation for inner security.
In Letter (4 Mar 1950), replying to a grieving father over the loss of a young son. In Dear Professor Einstein: Albert Einstein’s Letters to and from Children (2002), 184.
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Archimedes constructing his circle pays with his life for his defective biological adaptation to immediate circumstances.
In Ernst Mach and Thomas J. McCormack, Knowledge and Error: Sketches on the Psychology of Enquiry (1976, 2012), Chap. 4, 45. First published in German as Erkenntnis und Irrtum (1905). Translated from 5th edition (1926). The context is the legend that Archimedes was too engrossed in his work to pay attention to an invading Roman soldier, who killed him.
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As our circle of knowledge expands, so does the circumference of darkness surrounding it.
…...
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As to Bell's talking telegraph, it only creates interest in scientific circles, and, as a toy it is beautiful; but ... its commercial value will be limited.
Letter to William D. Baldwin, his attorney (1 Nov 1876). Telephone Investigating Committee, House of Representatives, United States 49th Congress, 1st Session, Miscellaneous Documents (1886), No. 355, 1186.
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Astronomy was thus the cradle of the natural sciences and the starting point of geometrical theories. The stars themselves gave rise to the concept of a ‘point’; triangles, quadrangles and other geometrical figures appeared in the constellations; the circle was realized by the disc of the sun and the moon. Thus in an essentially intuitive fashion the elements of geometrical thinking came into existence.
In George Edward Martin, The Foundations of Geometry and the Non-Euclidean Plane (1982), 72.
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Behind every man’s busy-ness there should be a level of undisturbed serenity and industry, as within the reef encircling a coral isle there is always an expanse of still water, where the depositions are going on which will finally raise it above the surface.
In A Week on the Concord and Merrimack Rivers (1862), 380.
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Circles to square and cubes to double
Would give a man excessive trouble.
The longitude uncertain roams,
In spite of Whiston and his bombs.
In 'Alma', Canto III, in Samuel Johnson, The Works of the English Poets, from Chaucer to Cowper (1810), 203. The reference to longitude reflects the difficulty of its determination at sea, and the public interest in the attempts to win the prize instituted by the British government in 1714 for a successful way to find longitude at sea (eventually won by John Harrison's chronometer). In this poem, William Whiston (who succeeded Isaac Newton as Lucasian Professor at Cambridge) is being satirized for what many thought was a crack-brained scheme to find the longitude. This proposed, with Humphrey Ditton, the use of widely separated ships firing off shells programmed to explode at a set time, and calculation of distance between them made from the time-lag between the observed sounds of the explosions using the known speed of sound.
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De Morgan was explaining to an actuary what was the chance that a certain proportion of some group of people would at the end of a given time be alive; and quoted the actuarial formula, involving p [pi], which, in answer to a question, he explained stood for the ratio of the circumference of a circle to its diameter. His acquaintance, who had so far listened to the explanation with interest, interrupted him and exclaimed, “My dear friend, that must be a delusion, what can a circle have to do with the number of people alive at a given time?”
In Mathematical Recreations and Problems (1896), 180; See also De Morgan’s Budget of Paradoxes (1872), 172.
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Dr. M.L. von Franz has explained the circle (or sphere) as a symbol of Self. It expresses the totality of the psyche in all its aspects, including the relationship between man and the whole of nature. It always points to the single most vital aspect of life, its ultimate wholeness.
In Aniela Jaffé, 'Symbolism in the Visual Arts', collected in Carl Jung (ed.), Man and His Symbols (1964, 1968), 266.
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Equations are Expressions of Arithmetical Computation, and properly have no place in Geometry, except as far as Quantities truly Geometrical (that is, Lines, Surfaces, Solids, and Proportions) may be said to be some equal to others. Multiplications, Divisions, and such sort of Computations, are newly received into Geometry, and that unwarily, and contrary to the first Design of this Science. For whosoever considers the Construction of a Problem by a right Line and a Circle, found out by the first Geometricians, will easily perceive that Geometry was invented that we might expeditiously avoid, by drawing Lines, the Tediousness of Computation. Therefore these two Sciences ought not to be confounded. The Ancients did so industriously distinguish them from one another, that they never introduced Arithmetical Terms into Geometry. And the Moderns, by confounding both, have lost the Simplicity in which all the Elegance of Geometry consists. Wherefore that is Arithmetically more simple which is determined by the more simple Equation, but that is Geometrically more simple which is determined by the more simple drawing of Lines; and in Geometry, that ought to be reckoned best which is geometrically most simple.
In 'On the Linear Construction of Equations', Universal Arithmetic (1769), Vol. 2, 470.
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Euler was a believer in God, downright and straightforward. The following story is told by Thiebault, in his Souvenirs de vingt ans de séjour à Berlin, … Thiebault says that he has no personal knowledge of the truth of the story, but that it was believed throughout the whole of the north of Europe. Diderot paid a visit to the Russian Court at the invitation of the Empress. He conversed very freely, and gave the younger members of the Court circle a good deal of lively atheism. The Empress was much amused, but some of her counsellors suggested that it might be desirable to check these expositions of doctrine. The Empress did not like to put a direct muzzle on her guest’s tongue, so the following plot was contrived. Diderot was informed that a learned mathematician was in possession of an algebraical demonstration of the existence of God, and would give it him before all the Court, if he desired to hear it. Diderot gladly consented: though the name of the mathematician is not given, it was Euler. He advanced toward Diderot, and said gravely, and in a tone of perfect conviction:
Monsieur, (a + bn) / n = x, donc Dieu existe; repondez!

Diderot, to whom algebra was Hebrew, was embarrassed and disconcerted; while peals of laughter rose on all sides. He asked permission to return to France at once, which was granted.
In Budget of Paradoxes (1878), 251. [The declaration in French expresses, “therefore God exists; please answer!” This Euler-Diderot anecdote, as embellished by De Morgan, is generally regarded as entirely fictional. Diderot before he became an encyclopedist was an accomplished mathematician and fully capable of recognizing—and responding to—the absurdity of an algebraic expression in proving the existence of God. See B.H. Brown, 'The Euler-Diderot Anecdote', The American Mathematical Monthly (May 1942), 49, No. 5, 392-303. —Webmaster.]
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Every progress that a church makes in the construction of its dogmas leads to a further taming of the free spirit; every new dogma … narrows the circle of free thought. … Science, on the other hand, liberates with every step of its development, it opens up new paths to thought … In other words, it allows the individual to be truly free.
Translated from the original German, “Jeder Fortschritt, den eine Kirche in dem Aufbau ihrer Dogmen macht, führt zu einer weiter gehenden Bändigung des freien Geistes; jedes neue Dogma … verengt den Kreis des freien Denkens. … Die Naturwissenschaft umgekehrt befreit mit jedem Schritte ihrer Entwicklung, sie eröffnet dem Gedanken neue Bahnen … Sie gestattet, mit anderen Worten, dem Einzelnen in vollem Masse wahr zu sein.” In Speech to the 24th meeting of the German Naturalists and Physicians at Rostock 'Ueber die Aufgaben der Naturwissenschaften in dem neuen nationalen Leben Deutschlands', (On the tasks of the natural sciences in the new national life of Germany), published in Chemisches Zentralblatt (11 Oct 1871), No. 41, 654-655. English version by Webmaster using Google translate.
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Finally, from what we now know about the cosmos, to think that all this was created for just one species among the tens of millions of species who live on one planet circling one of a couple of hundred billion stars that are located in one galaxy among hundreds of billions of galaxies, all of which are in one universe among perhaps an infinite number of universes all nestled within a grand cosmic multiverse, is provincially insular and anthropocentrically blinkered. Which is more likely? That the universe was designed just for us, or that we see the universe as having been designed just for us?
…...
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For many planet hunters, though, the ultimate goal is still greater (or actually, smaller) prey: terrestrial planets, like Earth, circling a star like the Sun. Astronomers already know that three such planets orbit at least one pulsar. But planet hunters will not rest until they are in sight of a small blue world, warm and wet, in whose azure skies and upon whose wind-whipped oceans shines a bright yellow star like our own.
…...
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Four circles to the kissing come,
The smaller are the benter.
The bend is just the inverse of
The distance from the centre.
Though their intrigue left Euclid dumb
There’s now no need for rule of thumb.
Since zero bend’s a dead straight line
And concave bends have minus sign,
The sum of squares of all four bends
Is half the square of their sum.
In poem, 'The Kiss Precise', Nature (20 Jun 1936), 137, 1021, as quoted, cited, explained and illustrated in Martin Gardner, The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems (2001), 139-141.
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I have said that mathematics is the oldest of the sciences; a glance at its more recent history will show that it has the energy of perpetual youth. The output of contributions to the advance of the science during the last century and more has been so enormous that it is difficult to say whether pride in the greatness of achievement in this subject, or despair at his inability to cope with the multiplicity of its detailed developments, should be the dominant feeling of the mathematician. Few people outside of the small circle of mathematical specialists have any idea of the vast growth of mathematical literature. The Royal Society Catalogue contains a list of nearly thirty- nine thousand papers on subjects of Pure Mathematics alone, which have appeared in seven hundred serials during the nineteenth century. This represents only a portion of the total output, the very large number of treatises, dissertations, and monographs published during the century being omitted.
In Presidential Address British Association for the Advancement of Science, Section A, (1910), Nature, 84, 285.
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If any one should ask me what I consider the most distinctive, progressive feature of California, I should answer promptly, its cable-car system. And it is not alone its system which seems to have reached a point of perfection, but the amazing length of the ride that is given you for the chink of a nickel. I have circled this city of San Francisco, … for this smallest of Southern coins.
In Letters from California (1888), 33.
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If there is an underlying oneness of all things, it does not matter where we begin, whether with stars, or laws of supply and demand, or frogs, or Napoleon Bonaparte. One measures a circle, beginning anywhere.
Lo! (1931, 1941), 8.
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If they would, for Example, praise the Beauty of a Woman, or any other Animal, they describe it by Rhombs, Circles, Parallelograms, Ellipses, and other geometrical terms …
In 'A Voyage to Laputa', Travels Into Several Remote Nations of the World by Captain Lemuel Gulliver (1726), Vol 2, Part 3, 26. (Gulliver’s Travels)
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In due time the evolution theory will have to abate its vehemence, cannot be allow’d to dominate everything else, and will have to take its place as a segment of the circle, the cluster—as but one of many theories, many thoughts, of profoundest value—and readjusting the differentiating much, yet leaving the divine secrets just as inexplicable and unreachable as before—maybe more so.
In Specimen days & Collect (1883), 326.
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In the human body the central point is naturally the navel. For if a man be placed flat on his back, with his hands and feet extended, and a pair of compasses centered at his navel, the fingers and toes of his two hands and feet will touch the circumference of a circle described therefrom.
Vitruvius
Leonardo da Vinci’s depiction of a man with outstretched limbs inscribed in a circle is thus called the Vitruvian Man (c. 1490). In De Architectura, Book 3, Chap 1, Sec. 3. As translated in Morris Hicky Morgan (trans.), Vitruvius: The Ten Books on Architecture (1914), 73.
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Indeed, we need not look back half a century to times which many now living remember well, and see the wonderful advances in the sciences and arts which have been made within that period. Some of these have rendered the elements themselves subservient to the purposes of man, have harnessed them to the yoke of his labors and effected the great blessings of moderating his own, of accomplishing what was beyond his feeble force, and extending the comforts of life to a much enlarged circle, to those who had before known its necessaries only.
From paper 'Report of the Commissioners Appointed to Fix the Site of the University of Virginia' (Dec 1818), reprinted in Annual Report of the Board of Visitors of the University of Virginia for the Fiscal Year Ending May 31, 1879 (1879), 10. Collected in Commonwealth of Virginia, Annual Reports of Officers, Boards, and Institutions of the Commonwealth of Virginia, for the Year Ending September 30, 1879 (1879).
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It is admitted, on all hands, that the Scriptures are not intended to resolve physical questions, or to explain matters in no way related to the morality of human actions; and if, in consequence of this principle, a considerable latitude of interpretation were not allowed, we should continue at this moment to believe, that the earth is flat; that the sun moves round the earth; and that the circumference of a circle is no more than three times its diameter.
In The Works of John Playfair: Vol. 1: Illustrations of the Huttonian Theory of the Earth (1822), 137.
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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.
In 'Prop. 47: The human mind possesses an adequate knowledge of the eternal and infinite essence of God', Ethic, translated by William Hale White (1883), 93-94. Collected in The English and Foreign Philosophical Library, Vol. 21.
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Mathematics accomplishes really nothing outside of the realm of magnitude; marvellous, however, is the skill with which it masters magnitude wherever it finds it. We recall at once the network of lines which it has spun about heavens and earth; the system of lines to which azimuth and altitude, declination and right ascension, longitude and latitude are referred; those abscissas and ordinates, tangents and normals, circles of curvature and evolutes; those trigonometric and logarithmic functions which have been prepared in advance and await application. A look at this apparatus is sufficient to show that mathematicians are not magicians, but that everything is accomplished by natural means; one is rather impressed by the multitude of skilful machines, numerous witnesses of a manifold and intensely active industry, admirably fitted for the acquisition of true and lasting treasures.
In Werke [Kehrbach] (1890), Bd. 5, 101. As quoted, cited and translated in Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath’s Quotation-Book (1914), 13.
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Mathematics will not be properly esteemed in wider circles until more than the a b c of it is taught in the schools, and until the unfortunate impression is gotten rid of that mathematics serves no other purpose in instruction than the formal training of the mind. The aim of mathematics is its content, its form is a secondary consideration and need not necessarily be that historic form which is due to the circumstance that mathematics took permanent shape under the influence of Greek logic.
In Die Entivickelung der Mathematik in den letzten Jahrhunderten (1884), 6.
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Mathematics … certainly would never have come into existence if mankind had known from the beginning that in all nature there is no perfectly straight line, no true circle, no standard of measurement.
From 'Of the First and Last Things', All Too Human: A Book For Free Spirits (1878, 1908), Part 1, section 11, 31.
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Mathematics, indeed, is the very example of brevity, whether it be in the shorthand rule of the circle, c = πd, or in that fruitful formula of analysis, e = -1, —a formula which fuses together four of the most important concepts of the science,—the logarithmic base, the transcendental ratio π, and the imaginary and negative units.
In 'The Poetry of Mathematics', The Mathematics Teacher (May 1926), 19, No. 5, 293.
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Meton: With the straight ruler I set to work
To make the circle four-cornered.
First(?) allusion to the problem of squaring the circle. As quoted in B. L. Van Der Waerden, Science Awakening (1954, 2012), Vol. 1, 130, Aristophanes writes this as a joke in his The Birds, which implies the scholarly problem of the quadrature of the circle was known to his audience in ancient Athens.
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Newton’s theory is the circle of generalization which includes all the others [as Kepler’s laws, Ptolemy’s theory, etc.];—the highest point of the inductive ascent;—the catastrophe of the philosophic drama to which Plato had prologized;— the point to which men’s minds had been journeying for two thousand years.
In History of the Inductive Sciences, Bk. 7, chap. 2, sect. 5.
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Science … is perpetually advancing. It is like a torch in the sombre forest of mystery. Man enlarges every day the circle of light which spreads round him, but at the same time, and in virtue of his very advance, he finds himself confronting, at an increasing number of points, the darkness of the Unknown.
In Einstein and the Universe; A Popular Exposition of the Famous Theory (1922), xvi.
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Some think that the earth remains at rest. But Philolaus the Pythagorean believes that, like the sun and moon, it revolves around the fire in an oblique circle. Heraclides of Pontus, and Ephantus the Pythagorean make the earth move, not in a progressive motion, but like a wheel in a rotation from west to east about its own center.
From Preface to Book on the Revolutions.
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Starres by the Sun are not inlarg’d but showne.
Gentle love deeds, as blossomes on a bough,
From loves awaken’d root doe bud out now.
If, as in water stir’d more circles bee
Produc’d by one, love such additions take,
Those like to many spheares, but one heaven make,
For, they are all concentrique unto thee.
From poem 'Loves Growth'in Poems on Several Occasions (1719), 23-24.
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Stay your rude steps, or e’er your feet invade
The Muses’ haunts,ye sons of War and Trade!
Nor you, ye legion fiends of Church and Law,
Pollute these pages with unhallow’d paw!
Debased, corrupted, grovelling, and confin’d,
No definitions touch your senseless mind;
To you no Postulates prefer their claim,
No ardent Axioms your dull souls inflame;
For you no Tangents touch, no Angles meet,
No Circles join in osculation sweet!
From poem, with co-authors John Hookham Frere, George Canning and George Ellis, The Loves of the Triangles: A Mathematical and Philosophical Poem, Canto I, collected in Poetry of the Anti-Jacobin (1854), 124.
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The chemist works along his own brilliant line of discovery and exposition; the astronomer has his special field to explore; the geologist has a well-defined sphere to occupy. It is manifest, however, that not one of these men can tell the whole tale, and make a complete story of creation. Another man is wanted. A man who, though not necessarily going into formal science, sees the whole idea, and speaks of it in its unity. This man is the theologian. He is not a chemist, an astronomer, a geologist, a botanist——he is more: he speaks of circles, not of segments; of principles, not of facts; of causes and purposes rather than of effects and appearances. Not that the latter are excluded from his study, but that they are so wisely included in it as to be put in their proper places.
In The People's Bible: Discourses Upon Holy Scripture: Vol. 1. Genesis (1885), 120.
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The description of right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn.
From Principia Mathematica, Book 1, in Author’s Preface to the translation from the Latin by ‎Andrew Motte, as The Mathematical Principles of Natural Philosophy (1729), Vol. 1, second unpaginated page of the Preface.
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The generalizations of science sweep on in ever-widening circles, and more aspiring flights, through a limitless creation.
In 'The Darwinian Hypothesis: Darwin on the Origin of Species', Man's Place in Nature and Other Essays (1910), 337.
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The greater is the circle of light, the greater is the boundary of the darkness by which it is confined. But, notwithstanding this, the more light get, the more thankful we ought to be, for by this means we have the greater range for satisfactory contemplation. time the bounds of light will be still farther extended; and from the infinity of the divine nature, and the divine works, we may promise ourselves an endless progress in our investigation them: a prospect truly sublime and glorious.
Experiments and Observations with a Continuation of the Observations on Air (1781), Vol. 2, ix.
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The skein of human continuity must often become this tenuous across the centuries (hanging by a thread, in the old cliche’), but the circle remains unbroken if I can touch the ink of Lavoisier’s own name, written by his own hand. A candle of light, nurtured by the oxygen of his greatest discovery, never burns out if we cherish the intellectual heritage of such unfractured filiation across the ages. We may also wish to contemplate the genuine physical thread of nucleic acid that ties each of us to the common bacterial ancestor of all living creatures, born on Lavoisier’s ancienne terre more than 3.5 billion years ago– and never since disrupted, not for one moment, not for one generation. Such a legacy must be worth preserving from all the guillotines of our folly.
…...
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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.
In Johann Friedrich Herbart, Henry M. Felkin (trans.) and Emmie Felkin (trans.), Letters and Lectures on Education [Felkin] (1898), 117.
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Time has a different quality in a forest, a different kind of flow. Time moves in circles, and events are linked, even if it’s not obvious that they are linked. Events in a forest occur with precision in the flow of tree time, like the motions of an endless dance.
The Wild Trees: A Story of Passion and Daring
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Vision, in my view, is the cause of the greatest benefit to us, inasmuch as none of the accounts now given concerning the Universe would ever have been given if men had not seen the stars or the sun or the heavens. But as it is, the vision of day and night and of months and circling years has created the art of number and has given us not only the notion of Time but also means of research into the nature of the Universe. From these we have procured Philosophy in all its range, than which no greater boon ever has come or will come, by divine bestowal, unto the race of mortals.
Plato
…...
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Voice is a flowing breath of air, perceptible to the hearing by contact. It moves in an endless number of circular rounds, like the innumerably increasing circular waves which appear when a stone is thrown into smooth water, and which keep on spreading indefinitely from the centre.
Vitruvius
In De Architectura, Book 5, Chap 1, Sec. 6. As translated in Morris Hicky Morgan (trans.), Vitruvius: The Ten Books on Architecture (1914), 138.
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What distinguishes the straight line and circle more than anything else, and properly separates them for the purpose of elementary geometry? Their self-similarity. Every inch of a straight line coincides with every other inch, and of a circle with every other of the same circle. Where, then, did Euclid fail? In not introducing the third curve, which has the same property—the screw. The right line, the circle, the screw—the representations of translation, rotation, and the two combined—ought to have been the instruments of geometry. With a screw we should never have heard of the impossibility of trisecting an angle, squaring the circle, etc.
From Letter (15 Feb 1852) to W.R. Hamilton, collected in Robert Perceval Graves, Life of W.R. Hamilton (1889), Vol. 3, 343.
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When intersected by a plane, the sphere displays in this section the circle, the genuine image of the created mind, placed in command of the body which it is appointed to rule; and this circle is to the sphere as the human mind is to the Mind Divine.
As quoted in Wolfgang Pauli, 'The Influence of Archetypal Ideas on the Scientific Theories of Kepler', as translated and collected in Writings on Physics and Philosophy (1994), 225. With Latin from Harmonia Mundi, Liber IV, Caput 1, collected in Christian Frisch (ed.), Opera Omnia (1864), Vol. 5, 223: “ plano vero sectum sphaericum circulum sectione repraesentat, mentis creatae, quae corpori regendo sit praefecta, genuinam imaginem, quae in ea proportione sit ad sphaericum, ut est mens humana ad divinam,”
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When the child outgrows the narrow circle of family life … then comes the period of the school, whose object is to initiate him into the technicalities of intercommunication with his fellow-men, and to familiarize him with the ideas that underlie his civilization, and which he must use as tools of thought if he would observe and understand the phases of human life around him; for these … are invisible to the human being who has not the aid of elementary ideas with which to see them.
In Psychologic Foundations of Education: An Attempt to Show the Genesis of the Higher Faculties of the Mind (1907), 265.
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Why is geometry often described as “cold” and “dry?” One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line… Nature exhibits not simply a higher degree but an altogether different level of complexity.
From The Fractal Geometry of Nature (1977, 1983), Introduction, xiii.
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Yet the widespread [planetary theories], advanced by Ptolemy and most other [astronomers], although consistent with the numerical [data], seemed likewise to present no small difficulty. For these theories were not adequate unless they also conceived certain equalizing circles, which made the planet appear to move at all times with uniform velocity neither on its deferent sphere nor about its own [epicycle's] center … Therefore, having become aware of these [defects], I often considered whether there could perhaps be found a more reasonable arrangement of circles, from which every apparent irregularity would be derived while everything in itself would move uniformly, as is required by the rule of perfect motion.
From Nicholaus Copernicus, Edward Rosen (trans.), Pawel Czartoryski (ed.) 'Commentariolus', in Nicholas Copernicus: Minor Works (1985), 81-83. Excerpted in Lisa M. Dolling, Arthur F. Gianelli and Glenn N. Statile (eds.) The Tests of Time: Readings in the Development of Physical Theory (2003), 40.
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[As a youth, fiddling in my home laboratory] I discovered a formula for the frequency of a resonant circuit which was 2π x sqrt(LC) where L is the inductance and C the capacitance of the circuit. And there was π, and where was the circle? … I still don’t quite know where that circle is, where that π comes from.
From address to the National Science Teachers’ Association convention (Apr 1966), 'What Is Science?', collected in Richard Phillips Feynman and Jeffrey Robbins (ed.), The Pleasure of Finding Things Out: The Best Short Works of Richard P. Feynman (1999, 2005), 177.
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[Wolfgang Bolyai] was extremely modest. No monument, said he, should stand over his grave, only an apple-tree, in memory of the three apples: the two of Eve and Paris, which made hell out of earth, and that of Newton, which elevated the earth again into the circle of the heavenly bodies.
In History of Elementary Mathematics (1910), 273.
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Carl Sagan Thumbnail In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time something like that happened in politics or religion. (1987) -- Carl Sagan
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