Euclid Quotes

Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.

Reading list (1972 edition)[edit] 1. Homer – Iliad, Odyssey 2. The Old Testament 3. Aeschylus – Tragedies 4. Sophocles – Tragedies 5. Herodotus – Histories 6. Euripides – Tragedies 7. Thucydides – History of the Peloponnesian War 8. Hippocrates – Medical Writings 9. Aristophanes – Comedies 10. Plato – Dialogues 11. Aristotle – Works 12. Epicurus – Letter to Herodotus; Letter to Menoecus 13. Euclid – Elements 14. Archimedes – Works 15. Apollonius of Perga – Conic Sections 16. Cicero – Works 17. Lucretius – On the Nature of Things 18. Virgil – Works 19. Horace – Works 20. Livy – History of Rome 21. Ovid – Works 22. Plutarch – Parallel Lives; Moralia 23. Tacitus – Histories; Annals; Agricola Germania 24. Nicomachus of Gerasa – Introduction to Arithmetic 25. Epictetus – Discourses; Encheiridion 26. Ptolemy – Almagest 27. Lucian – Works 28. Marcus Aurelius – Meditations 29. Galen – On the Natural Faculties 30. The New Testament 31. Plotinus – The Enneads 32. St. Augustine – On the Teacher; Confessions; City of God; On Christian Doctrine 33. The Song of Roland 34. The Nibelungenlied 35. The Saga of Burnt Njál 36. St. Thomas Aquinas – Summa Theologica 37. Dante Alighieri – The Divine Comedy;The New Life; On Monarchy 38. Geoffrey Chaucer – Troilus and Criseyde; The Canterbury Tales 39. Leonardo da Vinci – Notebooks 40. Niccolò Machiavelli – The Prince; Discourses on the First Ten Books of Livy 41. Desiderius Erasmus – The Praise of Folly 42. Nicolaus Copernicus – On the Revolutions of the Heavenly Spheres 43. Thomas More – Utopia 44. Martin Luther – Table Talk; Three Treatises 45. François Rabelais – Gargantua and Pantagruel 46. John Calvin – Institutes of the Christian Religion 47. Michel de Montaigne – Essays 48. William Gilbert – On the Loadstone and Magnetic Bodies 49. Miguel de Cervantes – Don Quixote 50. Edmund Spenser – Prothalamion; The Faerie Queene 51. Francis Bacon – Essays; Advancement of Learning; Novum Organum, New Atlantis 52. William Shakespeare – Poetry and Plays 53. Galileo Galilei – Starry Messenger; Dialogues Concerning Two New Sciences 54. Johannes Kepler – Epitome of Copernican Astronomy; Concerning the Harmonies of the World 55. William Harvey – On the Motion of the Heart and Blood in Animals; On the Circulation of the Blood; On the Generation of Animals 56. Thomas Hobbes – Leviathan 57. René Descartes – Rules for the Direction of the Mind; Discourse on the Method; Geometry; Meditations on First Philosophy 58. John Milton – Works 59. Molière – Comedies 60. Blaise Pascal – The Provincial Letters; Pensees; Scientific Treatises 61. Christiaan Huygens – Treatise on Light 62. Benedict de Spinoza – Ethics 63. John Locke – Letter Concerning Toleration; Of Civil Government; Essay Concerning Human Understanding;Thoughts Concerning Education 64. Jean Baptiste Racine – Tragedies 65. Isaac Newton – Mathematical Principles of Natural Philosophy; Optics 66. Gottfried Wilhelm Leibniz – Discourse on Metaphysics; New Essays Concerning Human Understanding;Monadology 67. Daniel Defoe – Robinson Crusoe 68. Jonathan Swift – A Tale of a Tub; Journal to Stella; Gulliver's Travels; A Modest Proposal 69. William Congreve – The Way of the World 70. George Berkeley – Principles of Human Knowledge 71. Alexander Pope – Essay on Criticism; Rape of the Lock; Essay on Man 72. Charles de Secondat, baron de Montesquieu – Persian Letters; Spirit of Laws 73. Voltaire – Letters on the English; Candide; Philosophical Dictionary 74. Henry Fielding – Joseph Andrews; Tom Jones 75. Samuel Johnson – The Vanity of Human Wishes; Dictionary; Rasselas; The Lives of the Poets

The laws of nature are but the mathematical thoughts of God.

There is no Royal Road to Geometry.

What has been affirmed without proof can also be denied without proof.

I would say, if you like, that the party is like an out-moded mathematics...that is to say, the mathematics of Euclid. We need to invent a non-Euclidian mathematics with respect to political discipline.

And so will I here state just plainly and briefly that I accept God. But I must point out one thing: if God does exist and really created the world, as we well know, he created it according to the principles of Euclidean geometry and made the human brain capable of grasping only three dimensions of space. Yet there have been and still are mathematicians and philosophers-among them some of the most outstanding-who doubt that the whole universe or, to put it more generally, all existence was created to fit Euclidean geometry; they even dare to conceive that two parallel lines that, according to Euclid, never do meet on earth do, in fact, meet somewhere in infinity. And so my dear boy, I’ve decided that I am incapable of understanding of even that much, I cannot possibly understand about God.

In other words, to put it into Euclid, or old-fashioned plane geometry, a straight line is not the shortest distance between two points.

I do not think there is a demonstrative proof (like Euclid) of Christianity, nor of the existence of matter, nor of the good will and honesty of my best and oldest friends. I think all three are (except perhaps the second) far more probable than the alternatives. The case for Christianity in general is well given by Chesterton…As to why God doesn't make it demonstratively clear; are we sure that He is even interested in the kind of Theism which would be a compelled logical assent to a conclusive argument? Are we interested in it in personal matters? I demand from my friend trust in my good faith which is certain without demonstrative proof. It wouldn't be confidence at all if he waited for rigorous proof. Hang it all, the very fairy-tales embody the truth. Othello believed in Desdemona's innocence when it was proved: but that was too late. Lear believed in Cordelia's love when it was proved: but that was too late. 'His praise is lost who stays till all commend.' The magnanimity, the generosity which will trust on a reasonable probability, is required of us. But supposing one believed and was wrong after all? Why, then you would have paid the universe a compliment it doesn't deserve. Your error would even so be more interesting and important than the reality. And yet how could that be? How could an idiotic universe have produced creatures whose mere dreams are so much stronger, better, subtler than itself?

I suppose I thought your mind would be full of dresses and dances." "Well, it's not. It's full of Socrates and Euclid.

Handwriting is a spiritual designing, even though it appears by means of a material instrument.

The Ottoman Turks were about to capture Constantinople, unleashing on Italy a migration of fleeing scholars with bundles of manuscripts containing the ancient wisdom of Euclid, Ptolemy, Plato, and Aristotle.

He thought that when he had healed sufficiently, and withdrawn from the capital, he might write the magus a letter and open a correspondence on Euclid, or Thales, or the new idea from the north, that the sun and not the Earth might be the centre of the universe.

[it was not a] circle—just a concrete platform with a pay phone and a sign that read EUCLID CIRCLE. I thought Euclid would have been mad. “That’s so typical of your attitude,” Svetlana said. “You always think everyone is angry. Try to have some perspective. It’s over two thousand years after his death, he’s in Boston for the first time, they’ve named something after him—why should his first reaction be to get pissed off?

Development of Western science is based on two great achievements: the invention of the formal logical system (in Euclidean geometry) by the Greek philosophers, and the discovery of the possibility to find out causal relationships by systematic experiment (during the Renaissance). In my opinion, one has not to be astonished that the Chinese sages have not made these steps. The astonishing thing is that these discoveries were made at all.

…if geometry were as much opposed to our passions and present interests as is ethics, we should contest it and violate I but little less, notwithstanding all the demonstrations of Euclid and Archimedes…

My task is to explain to you as quickly as possible my essence, that is, what sort of man I am, what I believe in, and what I hope for, is that right? And therefore I declare that I accept God pure and simple. But this, however, needs to be noted: if God exists and if he indeed created the earth, then, as we know perfectly well, he created it in accordance with Euclidean geometry, and he created human reason with a conception of only three dimensions of space. At the same time there were and are even now geometers and philosophers, even some of the most outstanding among them, who doubt that the whole universe, or, even more broadly, the whole of being, was created purely in accordance with Euclidean geometry; they even dare to dream that two parallel lines, which according to Euclid cannot possibly meet on earth, may perhaps meet somewhere in infinity. I, my dear, have come to the conclusion that if I cannot understand even that, then it is not for me to understand about God. I humbly confess that I do not have any ability to resolve such questions, I have a Euclidean mind, an earthly mind, and therefore it is not for us to resolve things that are not of this world. And I advise you never to think about it, Alyosha my friend, and most especially about whether God exists or not. All such questions are completely unsuitable to a mind created with a concept of only three dimensions. And so, I accept God, not only willingly, but moreover I also accept his wisdom and his purpose, which are completely unknown to us; I believe in order, in the meaning of life, I believe in eternal harmony, in which we are all supposed to merge, I believe in the Word for whom the universe is yearning, and who himself was 'with God,' who himself is God, and so on and so forth, to infinity. Many words have been invented on the subject. It seems I'm already on a good path, eh? And now imagine that in the final outcome I do not accept this world of God's, created by God, that I do not accept and cannot agree to accept. With one reservation: I have a childlike conviction that the sufferings will be healed and smoothed over, that the whole offensive comedy of human contradictions will disappear like a pitiful mirage, a vile concoction of man's Euclidean mind, feeble and puny as an atom, and that ultimately, at the world's finale, in the moment of eternal harmony, there will occur and be revealed something so precious that it will suffice for all hearts, to allay all indignation, to redeem all human villainy, all bloodshed; it will suffice not only to make forgiveness possible, but also to justify everything that has happened with men--let this, let all of this come true and be revealed, but I do not accept it and do not want to accept it! Let the parallel lines even meet before my own eyes: I shall look and say, yes, they meet, and still I will not accept it.

His hair, from much running of fingers through it, radiates in all directions and surrounds his head like a halo of glory, or like the second Corollary of Euclid I. 32.

Mathematicians have, according to Wright, been "unreasonably successful" in finding applications to apparently useless theorems, and often years after the theorems were first discovered.

It was in Alexandria that the circumference of the earth was first measured, the sun fixed at the center of the solar system, the workings of the brain and the pulse illuminated, the foundations of anatomy and physiology established, the definitive editions of Homer produced. It was in Alexandria that Euclid had codified geometry.

... in fact any good mind properly taught can think like Euclid and like Walt Whitman. The Renaissance, as we saw, was full of such minds, equally competent as poet and as engineers. The modern notion of "the two cultures," incompatible under one skull, comes solely from the proliferation of specialties in science; but these also divide scientists into groups that do not understand one another, the cause being the sheer mass of detail and the diverse terminologies. In essence the human mind remains one, not 2 or 60 different organs.

A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.

There was no hope for him this time: it was the third stroke. Night after night I had passed the house (it was vacation time) and studied the lighted square of window: and night after night I had found it lighted in the same way, faintly and evenly. If he was dead, I thought, I would see the reflection of candles on the darkened blind, for I knew that two candles must be set at the head of a corpse. He had often said to me: I am not long for this world and I had thought his words idle. Now I knew they were true. Every night as I gazed up at the window I said softly to myself the word paralysis. It had always sounded strangely in my ears, like the word gnomon in the Euclid and the word simony in the Catechism. But now it sounded to me like the name of some maleficent and sinful being. It filled me with fear, and yet I longed to be nearer to it and to look upon its deadly work.

It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It only gives nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relationship of these presumptions is left in the dark; one sees neither whether and in how far their connection is necessary, nor a priori whether it is possible. From Euclid to Legendre, to name the most renowned of modern writers on geometry, this darkness has been lifted neither by the mathematicians nor the philosophers who have laboured upon it.

Reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.

Spinoza is not to be read, he is to be studied; you must approach him as you would approach Euclid, recognizing that in these brief two hundred pages a man has written down his lifetime's thought with stoic sculptury of everything superfluous.

Do not try the parallels in that way: I know that way all along. I have measured that bottomless night, and all the light and all the joy of my life went out there. [Having himself spent a lifetime unsuccessfully trying to prove Euclid's postulate that parallel lines do not meet, Farkas discouraged his son János from any further attempt.]

Those two axioms are solid enough from a sociological perspective … but you rattled them off so quickly, like you’d already worked them out,” Luo Ji said, a little surprised. “I’ve been thinking about this for most of my life, but I’ve never spoken about it with anyone before. I don’t know why, really.… One more thing: To derive a basic picture of cosmic sociology from these two axioms, you need two other important concepts: chains of suspicion, and the technological explosion.” “Interesting terms. Can you explain them?” Ye Wenjie glanced at her watch. “There’s no time. But you’re clever enough to figure them out. Use those two axioms as a starting point for your discipline, and you might end up becoming the Euclid of cosmic sociology.” “I’m no Euclid. But I’ll remember what you said and give it a whirl. I might come to you for guidance, though.” “I’m afraid there won’t be that opportunity.… In that case, you might as well just forget I said anything. Either way, I’ve fulfilled my duty. Well, Xiao Luo, I’ve got to go.” “Take care, Professor.” Ye Wenjie went off through the twilight to her final meet-up. The

1. An 'unit' is that by virtue of which each of the things that exist is called one. 2. A 'number' is a multiple composed of units.

When a king asked Euclid, the mathematician, whether he could not explain his art to him in a more compendious manner? he was answered, that there was no royal way to geometry.

At the age of eleven, I began Euclid, with my brother as my tutor. ... I had not imagined that there was anything so delicious in the world. After I had learned the fifth proposition, my brother told me that it was generally considered difficult, but I had found no difficulty whatsoever. This was the first time it had dawned on me that I might have some intelligence.

Believe me, if Archimedes ever had the grand entrance of a girl as pretty as Gloria to look forward to, he would never have spent so much time calculating the value of Pi. He would have been baking her a Pie! If Euclid had ever beheld a vision of loveliness like the one I see walking into my anti-math class, he would have forgotten all the geometry of lines and planes, and concentrated on the sweet simplicity of soft curves. If Pythagoras had ever had a girl look at him the way Gloria's eyes fix in my direction, he would have given up his calculations on the hypotenuse of right triangles and run for the hills to pick a bouquet of wildflowers.

It was possible, I knew, to live on two planes at once - to have one's feet planted in reality but pointed in the direction of progress. It was what I had done as a kid on Euclid Avenue, what my family - and marginalized people more generally - had always done. You get somewhere by building that better reality, if at first only in your own mind. Or as Barack had put it that night, you may live in the world as it is, but you can still work to create the world as it should be.

Is it possible that the Pentateuch could not have been written by uninspired men? that the assistance of God was necessary to produce these books? Is it possible that Galilei ascertained the mechanical principles of 'Virtual Velocity,' the laws of falling bodies and of all motion; that Copernicus ascertained the true position of the earth and accounted for all celestial phenomena; that Kepler discovered his three laws—discoveries of such importance that the 8th of May, 1618, may be called the birth-day of modern science; that Newton gave to the world the Method of Fluxions, the Theory of Universal Gravitation, and the Decomposition of Light; that Euclid, Cavalieri, Descartes, and Leibniz, almost completed the science of mathematics; that all the discoveries in optics, hydrostatics, pneumatics and chemistry, the experiments, discoveries, and inventions of Galvani, Volta, Franklin and Morse, of Trevithick, Watt and Fulton and of all the pioneers of progress—that all this was accomplished by uninspired men, while the writer of the Pentateuch was directed and inspired by an infinite God? Is it possible that the codes of China, India, Egypt, Greece and Rome were made by man, and that the laws recorded in the Pentateuch were alone given by God? Is it possible that Æschylus and Shakespeare, Burns, and Beranger, Goethe and Schiller, and all the poets of the world, and all their wondrous tragedies and songs are but the work of men, while no intelligence except the infinite God could be the author of the Pentateuch? Is it possible that of all the books that crowd the libraries of the world, the books of science, fiction, history and song, that all save only one, have been produced by man? Is it possible that of all these, the bible only is the work of God?

Old Euclid drew a circle On a sand-beach long ago. He bounded and enclosed it With angles thus and so. His set of solemn greybeards Nodded and argued much Of arc and circumference, Diameter and such. A silent child stood by them From morning until noon Because they drew such charming Round pictures of the moon.

I then began to study arithmetical questions without any great apparent result, and without suspecting that they could have the least connexion with my previous researches. Disgusted at my want of success, I went away to spend a few days at the seaside, and thought of entirely different things. One day, as I was walking on the cliff, the idea came to me, again with the same characteristics of conciseness, suddenness, and immediate certainty, that arithmetical transformations of indefinite ternary quadratic forms are identical with those of non-Euclidian geometry.

I am trying to explain as quickly as possible my essential nature, that is, what manner of man I am, what I believe in, and for what I hope, that's it, isn't it? And therefore I tell you that I accept God honestly and simply. But you must note this: If God exists and if He really did create the world, then, as we all know, He created it according to the geometry of only three dimensions in space. Yet there have been some very distinguished ones, who doubt whether the whole universe, or to speak more generally the whole of being, was only created in Euclid's geometry; they even dare to dream that two parallel lines, which according to Euclid can never meet on earth, may meet somewhere in infinity. I have come to the conclusion that, since I can't understand even that, I can't expect to understand about God. I acknowledge humbly that I have no faculty for settling such questions, I have a Euclidian earthly mind, and how could I solve problems that are not of this world? And I advise you never to think about it either, my dear Alyosha, especially about God, whether He exists or not. All such questions are utterly inappropriate for a mind created with a conception of only three dimensions. And so I accept God and am glad to, and what's more I accept His wisdom, His purpose - which are utterly beyond our ken; I believe in the underlying order and the meaning of life; I believe in the eternal harmony in which they say we shall one day be blended. I believe in the Word to Which the universe is striving, and Which Itself was "with God", and Which Itself is God and so on, and so on, to infinity.

ὅπερ ἔδει δεῖξαι

His conclusions were as infallible as so many propositions of Euclid. So startling would his results appear to the uninitiated that until they learned the processes by which he had arrived at them they might well consider him as a necromancer.

After the death of Archimedes in 212 BCE, the topic of motion was effectively abandoned; it did not resurface for another 1,400 years, when Gerard of Brussels revived the mathematical works of Euclid and Archimedes and came very close to defining speed as a ratio of distance to time.

The proof is by reductio ad absurdum, and reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons5. It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.

As to the need of improvement there can be no question whilst the reign of Euclid continues. My own idea of a useful course is to begin with arithmetic, and then not Euclid but algebra. Next, not Euclid, but practical geometry, solid as well as plane; not demonstration, but to make acquaintance. Then not Euclid, but elementary vectors, conjoined with algebra, and applied to geometry. Addition first; then the scalar product. Elementary calculus should go on simultaneously, and come into vector algebraic geometry after a bit. Euclid might be an extra course for learned men, like Homer...

As a monarch who should care more for the outlying colonies he knows on the map or through the report of his vicegerents, than for the trunk of his empire under his eyes at home, are we not more concerned about the shadowy life that we have in the hearts of others, and that portion in their thoughts and fancies which, in a certain far-away sense, belongs to us, than about the real knot of our identity - that central metropolis of self, of which alone we are immediately aware - or the diligent service of arteries and veins and infinitesimal activity of ganglia, which we know (as we know a proposition in Euclid) to be the source and substance of the whole?

In Leipzig [in the 14th century], the university found it necessary to promulgate a rule against throwing stones at the professors. As late as 1495, a German statute explicitly forbade anyone associated with the university from drenching freshmen with urine.

geognósticas

Euclid

A machine is as distinctively and brilliantly and expressively human as a violin sonata or a theorem in Euclid. —GREGORY VLASTOS

While Euclid himself may not have been the greatest mathematician who ever lived, he was certainly the greatest teacher of mathematics.

I have given up newspapers in exchange for Tacitus and Thucydides, for Newton and Euclid; and I find myself much the happier.

Mother Nature did not attend high school geometry courses or read the books of Euclid of Alexandria. Her geometry is jagged, but with a logic of its own and one that is easy to understand.

No very good sense can be given to the idea that the elements of Euclidean geometry may be found in nature because either everything is found in nature or nothing is. Euclidean geometry is a theory, and the elements of a theory may be interpreted only in terms demanded by the theory itself. Euclid’s axioms are satisfied in the Euclidean plane. Nature has nothing to do with it.

In the pentagram, the Pythagoreans found all proportions well-known in antiquity: arithmetic, geometric, harmonic, and also the well-known golden proportion, or the golden ratio. ... Probably owing to the perfect form and the wealth of mathematical forms, the pentagram was chosen by the Pythagoreans as their secret symbol and a symbol of health. - Alexander Voloshinov [As quoted in Stakhov]

You have attempted to tinge detection with romanticism, which produces much the same effect as if you worked a love-story or an elopement into the fifth proposition of Euclid." - Holmes to Watson, The Sign of Four

The one created thing which we cannot look at is the one thing in the light of which we look at everything. Like the sun at noonday, mysticism explains everything else by the blaze of its own victorious invisibility. Detached intellectualism is (in the exact sense of a popular phrase) all moonshine; for it is light without heat, and it is secondary light, reflected from a dead world. But the Greeks were right when they made Apollo the god both of imagination and of sanity; for he was both the patron of poetry and the patron of healing. Of necessary dogmas and a special creed I shall speak later. But that transcendentalism by which all men live has primarily much the position of the sun in the sky. We are conscious of it as of a kind of splendid confusion; it is something both shining and shapeless, at once a blaze and a blur. But the circle of the moon is as clear and unmistakable, as recurrent and inevitable, as the circle of Euclid on a blackboard. For the moon is utterly reasonable; and the moon is the mother of lunatics and has given to them all her name.

Optimism reigned in my family’s little apartment on Euclid Avenue. I saw it in my father, in the way he moved around as if nothing were wrong with his body, as if the disease that would someday take his life just didn’t exist.

On what may be the last page he wrote in his notebooks, Leonardo drew four right triangles with bases of differing lengths (fig. 143). Inside of each he fit a rectangle, and then he shaded the remaining areas of the triangle. In the center of the page he made a chart with boxes labeled with the letter of each rectangle, and below it he described what he was trying to accomplish. As he had done obsessively over the years, he was using the visualization of geometry to help him understand the transformation of shapes. Specifically, he was trying to understand the formula for keeping the area of a right triangle the same while varying the lengths of its two legs. He had fussed with this problem, explored by Euclid, repeatedly over the years. It was a puzzle that, by this point in his life, as he turned sixty-seven and his health faded, might seem unnecessary to solve. To anyone other than Leonardo, it may have been.

Women don’t use knives,’ Griffoni answered, reciting it as though she were Euclid listing another axiom. Although he agreed with her, Brunetti was curious about the basis for her belief. ‘You offering proof of that?’ ‘Kitchens,’ she said laconically. ‘Kitchens?’ ‘The knives are kept in the kitchen, and their husbands pass through there every day, countless times, yet very few of them get stabbed. That’s because women don’t use knives, and they don’t stab people.

At the Stourbridge Fair in 1663, at age twenty, he purchased a book on astrology, “out of a curiosity to see what there was in it.” He read it until he came to an illustration which he could not understand, because he was ignorant of trigonometry. So he purchased a book on trigonometry but soon found himself unable to follow the geometrical arguments. So he found a copy of Euclid’s Elements of Geometry, and began to read. Two years later he invented the differential calculus.

Einstein supposes that space is Euclidean where it is sufficiently remote from matter, but that the presence of matter causes it to become slightly non-Euclidean—the more matter there is in the neighborhood, the more space will depart from Euclid.

My whole image is a short step to the right of bondage porn. It's like I was begging you to do this to me before we even met, isn't it? Like I was just waiting for someone to belong to. I'm everything you think I am. I want everything you think I do. Explore me.

Captain West advanced to meet me, and before our outstretched hands touched, before his face broke from repose to greeting and the lips moved to speech, I got the first astonishing impact of his personality. Long, lean, in his face a touch of race I as yet could only sense, he was as cool as the day was cold, as poised as a king or emperor, as remote as the farthest fixed star, as neutral as a proposition of Euclid. And then, just ere our hands met, a twinkle of--oh--such distant and controlled geniality quickened the many tiny wrinkles in the corner of the eyes; the clear blue of the eyes was suffused by an almost colourful warmth; the face, too, seemed similarly to suffuse; the thin lips, harsh-set the instant before, were as gracious as Bernhardt's when she moulds sound into speech.

Detection is, or ought to be, an exact science, and should be treated in the same cold and unemotional manner. You have attempted to tinge it with romanticism, which produces much the same effect as if you worked a love-story or an elopement into the fifth proposition of Euclid.

Chemistry, for me, had stopped being such a source. It led to the heart of Matter, and Matter was our ally precisely because the Spirit, dear to Fascism, was our enemy; but, having reached the fourth year of Pure Chemistry, I could no longer ignore the fact that chemistry itself, or at least that which we were being administered, did not answer my questions. To prepare phenyl bromide according to Gatterman was amusing, even exhilarating, but not very different from following Artusi's recipes. Why in that particular way and not in another? After having been force fed in liceo the truths revealed by Fascist Doctrine, all revealed, unproven truths either bored me stiff or aroused my suspicion. Did chemistry theorems exist? No; therefore you had to go further, not be satisfied with the quia go back to the origins, to mathematics and physics. The origins of chemistry were ignoble, or at least equivocal: the dens of the alchemists, their abominable hodgepodge of ideas and language, their confessed interest in gold, their Levantine swindles typical of charlatans or magicians; instead, at the origin of physics lay the strenuous clarity of the West – Archimedes and Euclid.

If the ancients had been able to see it as I see it now, Mr. Palomar thinks, they would have thought they had projected their gaze into the heaven of Plato's ideas, or in the immaterial space of the postulates of Euclid; but instead, thanks to some misdirection or other, this sight has been granted to me, who fear it is too beautiful to be true, too gratifying to my imaginary universe to belong to the real world. But perhaps it is this same distrust of our senses that prevents us from feeling comfortable in the universe. Perhaps the first rule I must impose on myself is this: stick to what I see.

we must not forget that the restful experience of enjoyable beauty is not limited to the contemplation of sensible objects. We can experience it as well in the contemplation of purely intelligible objects—the contemplation of truths we understand. “Mathematics,” wrote Bertrand Russell, “rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere … without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music …” Or, as the poet Edna St. Vincent Millay wrote in the opening line of her sonnet on Euclid, “Euclid alone has looked on beauty bare.

Yet there have been and still are mathematicians and philosophers who doubt whether the whole universe, or to speak more widely, the whole of being, was only created in Euclid's geometry. They even dare to dream that two parallel lines, which according to Euclid can never meet on earth, may meet somewhere in infinity.

Pythagoras also established the principle of deductive reasoning, which is the step-by-step process of starting with self-evident axioms (such as “2 + 2 = 4”) to build toward a new conclusion or fact. Deductive reasoning was later refined by Euclid, and it formed the basis of mathematical thinking into medieval times and beyond.

Please give it up. Fear it no less than the sensual passion, because it, too, may take up all your time and deprive you of your health, peace of mind and happiness in life. [Having himself spent a lifetime unsuccessfully trying to prove Euclid's postulate that parallel lines do not meet, Farkas discouraged his son János from any further attempt.]

Now we can see what makes mathematics unique. Only in mathematics is there no significant correction-only extension. Once the Greeks had developed the deductive method, they were correct in what they did, correct for all time. Euclid was incomplete and his work has been extended enormously, but it has not had to be corrected. His theorems are, every one of them, valid to this day. Ptolemy may have developed an erroneous picture of the planetary system, but the system of trigonometry he worked out to help him with his calculations remains correct forever. Each great mathematician adds to what came previously, but nothing needs to be uprooted. Consequently, when we read a book like A History of Mathematics, we get the picture of a mounting structure, ever taller and broader and more beautiful and magnificent and with a foundation, moreover, that is as untainted and as functional now as it was when Thales worked out the first geometrical theorems nearly 26 centuries ago. Nothing pertaining to humanity becomes us so well as mathematics. There, and only there, do we touch the human mind at its peak.

Porque o que estas denunciam — no enterroado do chão, no desmantelo dos cerros quase desnudos, no contorcido dos leitos secos dos ribeirões efêmeros, no constrito das gargantas e no quase convulsivo de uma flora decídua embaralhada em esgalhos — é de algum modo o martírio da terra, brutalmente golpeada pelos elementos variáveis, distribuídos por todas as modalidades climáticas.

The Greek excellence in mathematics was largely a direct consequence of their passion for knowledge for its own sake, rather than merely for practical purposes. A story has it that when a student who learned one geometrical proposition with Euclid asked, "But what do I gain from this?" Euclid told his slave to give the boy a coin, so that the student would see an actual profit.

Now we can see what makes mathematics unique. Only in mathematics is there no significant correction—only extension. Once the Greeks had developed the deductive method, they were correct in what they did, correct for all time. Euclid was incomplete and his work has been extended enormously, but it has not had to be corrected. His theorems are, every one of them, valid to this day.

... I left Caen, where I was living, to go on a geological excursion under the auspices of the School of Mines. The incidents of the travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go to some place or other. At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had time, as upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for convenience sake, I verified the result at my leisure.

I wonder if you've ever considered how strange it is that the educational and character-shaping structures of our culture expose us but a single time in our lives to the ideas of Socrates, Plato, Euclid, Aristotle, Herodotus, Augustine, Machiavelli, Shakespeare, Descartes, Rousseau, Newton, Racine, Darwin, Kant, Kierkegaard, Tolstoy, Schopenhauer, Goethe, Freud, Marx, Einstein, and dozens of others of the same rank, but expose us annually, monthly, weekly, and even daily to the ideas of persons like Jesus, Moses, Muhammad, and Buddha. Why is it, do you think, that we need quarterly lectures on charity, while a single lecture on the laws of thermodynamics is presumed to last us a lifetime? Why is the meaning of Christmas judged to be so difficult of comprehension that we must hear a dozen explications of it, not once in a lifetime, but every single year, year after year after year?

Euclid's Elements has been for nearly twenty-two centuries the encouragement and guide of that scientific thought which is one thing with the progress of man from a worse to a better state. The encouragement; for it contained a body of knowledge that was really known and could be relied on, and that moreover was growing in extent and application. For even at the time this book was written—shortly after the foundation of the Alexandrian Museum—Mathematics was no longer the merely ideal science of the Platonic school, but had started on her career of conquest over the whole world of Phenomena. The guide; for the aim of every scientific student of every subject was to bring his knowledge of that subject into a form as perfect as that which geometry had attained. Far up on the great mountain of Truth, which all the sciences hope to scale, the foremost of that sacred sisterhood was seen, beckoning for the rest to follow her.

This picture of matter curving space and curvaceous space dictating how matter and light will move has several striking features. It brings the non-Euclidean geometries that we talked about in the last chapter out from the library of pure mathematics into the arena of science. The vast collection of geometries describing spaces that are not simply the flat space of Euclid are the ones that Einstein used to capture the possible structures of space distorted by the presence of mass and energy.

My reading has been lamentably desultory and immedthodical. Odd, out of the way, old English plays, and treatises, have supplied me with most of my notions, and ways of feeling. In everything that relates to science, I am a whole Encyclopaedia behind the rest of the world. I should have scarcely cut a figure among the franklins, or country gentlemen, in King John's days. I know less geography than a schoolboy of six weeks standing. To me a map of old Ortelius is as authentic as Arrowsmith. I do not know whereabout Africa merges into Asia, whether Ethiopia lie in one or other of those great divisions, nor can form the remotest, conjecture of the position of New South Wales, or Van Diemen's Land. Yet do I hold a correspondence with a very dear friend in the first named of these two Terrae Incognitae. I have no astronomy. I do not know where to look for the Bear or Charles' Wain, the place of any star, or the name of any of them at sight. I guess at Venus only by her brightness - and if the sun on some portentous morn were to make his first appearance in the west, I verily believe, that, while all the world were grasping in apprehension about me, I alone should stand unterrified, from sheer incuriosity and want of observation. Of history and chronology I possess some vague points, such as one cannot help picking up in the course of miscellaneous study, but I never deliberately sat down to a chronicle, even of my own country. I have most dim apprehensions of the four great monarchies, and sometimes the Assyrian, sometimes the Persian, floats as first in my fancy. I make the widest conjectures concerning Egypt, and her shepherd kings. My friend M., with great pains taking, got me to think I understood the first proposition in Euclid, but gave me over in despair at the second. I am entirely unacquainted with the modern languages, and, like a better man than myself, have 'small Latin and less Greek'. I am a stranger to the shapes and texture of the commonest trees, herbs, flowers - not from the circumstance of my being town-born - for I should have brought the same inobservant spirit into the world with me, had I first seen it, 'on Devon's leafy shores' - and am no less at a loss among purely town objects, tool, engines, mechanic processes. Not that I affect ignorance - but my head has not many mansions, nor spacious, and I have been obliged to fill it with such cabinet curiosities as it can hold without aching. I sometimes wonder how I have passed my probation with so little discredit in the world, as I have done, upon so meagre a stock. But the fact is, a man may do very well with a very little knowledge, and scarce be found out, in mixed company; everybody is so much more ready to produce his own, than to call for a display of your acquisitions. But in a tete-a-tete there is no shuffling. The truth will out. There is nothing which I dread so much, as the being left alone for a quarter of an hour with a sensible, well-informed man that does not know me.

For what is it you and I are trying to do now? What I'm trying to do is to attempt to explain to you as quickly as possible the most important thing about me, that is to say, what sort of man I am, what I believe in what I hope for - that's it isn't it? And that's why I declare that I accept God plainly and simply. But there's this that has to be said: if God really exists and if he really has created the world, then, as we all know, he created it in accordance with the Euclidean geometry, and he created the human mind with the conception of only the three dimensions of space. And yet there have been and there still are mathematicians and philosophers, some of them indeed men of extraordinary genius, who doubt whether the whole universe, or, to put it more wildly, all existence was created only according to Euclidean geometry and they even dare to dream that two parallel lines which, according to Euclid can never meet on earth, may meet somewhere in infinity. I, my dear chap, have come to the conclusion that if I can't understand even that, then how can I be expected to understand about God? I humbly admit that I have no abilities for settling such questions. And I advise you too, Aloysha, my friend, never to think about it, and least of all about whether there is a God or not. All these problems which are entirely unsuitable to a mind created with the idea of only three dimensions. And so I accept God, and I accept him not only without reluctance, but what's more, I accept his divine wisdom and his purpose- which are completely beyond our comprehension.

When he was seventy-four years old the Cretan novelist Nikos Kazantzakis began a book. He called it Report to Greco... Kazantzakis thought of himself as a soldier reporting to his commanding officer on a mortal mission—his life. ... Well, there is only one Report to Greco, but no true book... was ever anything else than a report. ... A true book is a report upon the mystery of existence... it speaks of the world, of our life in the world. Everything we have in the books on which our libraries are founded—Euclid's figures, Leonardo's notes, Newton's explanations, Cervantes' myth, Sappho's broken songs, the vast surge of Homer—everything is a report of one kind or another and the sum of all of them together is our little knowledge of our world and of ourselves. Call a book Das Kapital or The Voyage of the Beagle or Theory of Relativity or Alice in Wonderland or Moby-Dick, it is still what Kazantzakis called his book—it is still a "report" upon the "mystery of things." But if this is what a book is... then a library is an extraordinary thing. ... The existence of a library is, in itself, an assertion. ... It asserts that... all these different and dissimilar reports, these bits and pieces of experience, manuscripts in bottles, messages from long before, from deep within, from miles beyond, belonged together and might, if understood together, spell out the meaning which the mystery implies. ... The library, almost alone of the great monuments of civilization, stands taller now than it ever did before. The city... decays. The nation loses its grandeur... The university is not always certain what it is. But the library remains: a silent and enduring affirmation that the great Reports still speak, and not alone but somehow all together...

Indeed, much of Newton’s intellectual development can be attributed to this tension between rationalism and mysticism. At the Stourbridge Fair in 1663, at age twenty, he purchased a book on astrology, ‘out of a curiosity to see what there was in it.’ He read it until he came to an illustration which he could not understand, because he was ignorant of trigonometry. So he purchased a book on trigonometry but soon found himself unable to follow the geometrical arguments. So he found a copy of Euclid’s Elements of Geometry, and began to read. Two years later he invented the differential calculus.

The theory of the regular solids, which is set forth in the thirteenth book of Euclid, was, in Plato's day, a recent discovery; it was completed by Theaetetus, who appears as a very young man in the dialogue that bears his name. It was, according to tradition, he who first proved that there are only five kinds of regular solids, and discovered the octahedron and the icosahedron.4 The regular tetrahedron, octahedron, and icosahedron, have equilateral triangles for their faces; the dodecahedron has regular pentagons, and cannot therefore be constructed out of Plato's two triangles. For this reason he does not use it in connection with the four elements.

Note II.—From all that has been said above it is clear, that we, in many cases, perceive and form our general notions:—(1.) From particular things represented to our intellect fragmentarily, confusedly, and without order through our senses (II. xxix. Coroll.); I have settled to call such perceptions by the name of knowledge from the mere suggestions of experience.4 (2.) From symbols, e.g., from the fact of having read or heard certain words we remember things and form certain ideas concerning them, similar to those through which we imagine things (II. xviii. note). I shall call both these ways of regarding things knowledge of the first kind, opinion, or imagination. (3.) From the fact that we have notions common to all men, and adequate ideas of the properties of things (II. xxxviii. Coroll., xxxix. and Coroll. and xl.); this I call reason and knowledge of the second kind. Besides these two kinds of knowledge, there is, as I will hereafter show, a third kind of knowledge, which we will call intuition. This kind of knowledge proceeds from an adequate idea of the absolute essence of certain attributes of God to the adequate knowledge of the essence of things. I will illustrate all three kinds of knowledge by a single example. Three numbers are given for finding a fourth, which shall be to the third as the second is to the first. Tradesmen without hesitation multiply the second by the third, and divide the product by the first; either because they have not forgotten the rule which they received from a master without any proof, or because they have often made trial of it with simple numbers, or by virtue of the proof of the nineteenth proposition of the seventh book of Euclid, namely, in virtue of the general property of proportionals. But with very simple numbers there is no need of this. For instance, one, two, three, being given, everyone can see that the fourth proportional is six; and this is much clearer, because we infer the fourth number from an intuitive grasping of the ratio, which the first bears to the second.

En vista de que ni siquiera esto soy capaz de comprender, he decidido no intentar comprender a Dios. Confieso humildemente mi incapacidad para resolver estas cuestiones. En esencia, mi mentalidad es la de Euclides: una mentalidad terrestre. ¿Para qué intentar resolver cosas que no son de este mundo? Te aconsejo que no te tortures el cerebro tratando de resolver estas cuestiones, y menos aún el problema de la existencia de Dios. ¿Existe o no existe? Estos puntos están fuera del alcance de la inteligencia humana, que sólo tiene la noción de las tres dimensiones. Por eso yo admito sin razonar no sólo la existencia de Dios, sino también su sabiduría y su finalidad para nosotros incomprensible.

There have been and there still are mathematicians and philosophers, some of them indeed men of extraordinary genius, who doubt whether the whole universe, or, to put it more widely, all existence, was created only according to Euclidean geometry and they even dare to dream that wo parallel lines which, according to Euclid, can never meet on earth, may meet somewhere in infinity. I, my dear chap, have come to the conclusion that if I can’t understand even that, then how can I be expected to understand about God? I humbly admit that I have no abilities for settling such questions. I have a Euclidean, an earthly mind, and so how can I be expected to solve problems which are not of this world.

But in philosophy, he was closer to his contemporary Siddhartha Gautama Buddha (c. 560 – 480 B.C.). Both believed in reincarnation, possibly as an animal, so even an animal could be inhabited by what was once a human soul. Thus, both placed a high value on all life, opposing the common practice of animal sacrifice and preaching strict vegetarianism.

In Steven Spielberg’s film Lincoln, the screenwriter Tony Kushner has the great emancipator explain Euclid’s axiom in the context of a discussion on the equality of the races: “Euclid’s first common notion is this: Things which are equal to the same thing are equal to each other. That’s a rule of mathematical reasoning. It’s true because it works. Has done and always will do. In his book Euclid says this is self-evident. You see, there it is, even in that 2,000-year-old book of mechanical law it is a self-evident truth.” Although Lincoln never actually uttered those words, there is every reason to think that he would have made just such an argument because it’s precisely what is implied in his 1854 argument that A is interchangeable with B.

Pythagoras was a charismatic figure and a genius, but he was also a good self-promoter. In Egypt, he not only learned Egyptian geometry but became the first Greek to learn Egyptian hieroglyphics, and eventually became an Egyptian priest, or the equivalent, initiated into their sacred rites. This gave him access to all their mysteries, even to the secret rooms in their temples.

Estas lagoas mortas, segundo a bela etimologia indígena, demarcam obrigatória escala ao caminhante. Associando-se às cacimbas e “caldeirões”, em que se abre a pedra, são-lhe recurso único na viagem penosíssima. Verdadeiros oásis, têm contudo, não raro, um aspecto lúgubre: localizadas em depressões, entre colinas nuas, envoltas pelos mandacarus despidos e tristes, como espectros de árvores;

One morning during Lent in 415, Hypatia climbed into her chariot, some say outside her residence, some say on a street intending to ride home. Several hundred of Cyril's stooges, Christian monks from a desert monastery, swooped upon her, beat her, and dragged her to a church. Inside the church they stripped her naked and peeled away her flesh with either sharpened tiles or broken bits of pottery.

Today we know that there are other solar systems only tens of light years away. Had the Golden Age continued unabated, we might by now have sent probes exploring them. We might have landed on the moon in the year 969 instead of 1969. We might have an understanding of space and life that is unimaginable to us today. Instead, events occurred that would delay the progress begun by the Greeks by a millennium.

What can you prove about space? How do you know where you are? Can space be curved? How many dimensions are there? How does geometry explain the natural order and unity of the cosmos? These are the questions behind the five geometric revolutions of world history. It started with a little scheme hatched by Pythagoras: to employ mathematics as the abstract system of rules that can model the physical universe.

There have been and still are geometricians and philosophers, and even some of the most distinguished, who doubt whether the whole universe, or to speak more widely the whole of existence, was only created in Euclid's geometry; they even dare to dream that two parallel lines, which according to Euclid can never meet on earth, may meet somewhere in infinity. I have come to the conclusion that, since I can't understand even that, I can't expect to understand about God. I acknowledge humbly that I have no faculty for settling such questions, I have a Euclidean earthly mind, and how could I solve problems that are not of this world? And I advise you never to think about it either, my dear Alyosha, especially about God, whether He exists or not. All such questions are utterly inappropriate for a mind created with an idea of only three dimensions.

Some persons fancy that bias and counter-bias are favorable to the extraction of truth–that hot and partisan debate is the way to investigate. This is the theory of our atrocious legal procedure. But Logic puts its heel upon this suggestion. It irrefragably demonstrates that knowledge can only be furthered by the real desire for it, and that the methods of obstinacy, of authority and every mode of trying to reach a foregone conclusion, are absolutely of no value. These things are proved. The reader is at liberty to think so or not as long as the proof is not set forth, or as long as he refrains from examining it. Just so, he can preserve, if he likes, his freedom of opinion in regard to the propositions of geometry; only, in that case, if he takes a fancy to read Euclid, he will do well to skip whatever he finds with A, B, C, etc., for, if he reads attentively that disagreeable matter, the freedom of his opinion about geometry may unhappily be lost forever.

The influence of geometry upon philosophy and scientific method has been profound. Geometry, as established by the Greeks, starts with axioms which are (or are deemed to be) self-evident, and proceeds, by deductive reasoning, to arrive at theorems that are very far from self-evident. The axioms and theorems are held to be true of actual space, which is something given in experience. It thus appeared to be possible to discover things about the actual world by first noticing what is self-evident and then using deduction. This view influenced Plato and Kant, and most of the intermediate philosophers. When the Declaration of Independence says 'we hold these truths to be self-evident', it is modelling itself on Euclid. The eighteenth-century doctrine of natural rights is a search for Euclidean axioms in politics.8 The form of Newton's Principia, in spite of its admittedly empirical material, is entirely dominated by Euclid. Theology, in its exact scholastic forms, takes its style from the same source. Personal religion is derived from ecstasy, theology from mathematics; and both are to be found in Pythagoras.

Twenty-four centuries ago, a Greek man stood at the sea’s edge watching ships disappear in the distance. Aristotle must have passed much time there, quietly observing many vessels, for eventually he was struck by a peculiar thought. All ships seemed to vanish hull first, then masts and sails. He wondered, how could that be? On a flat earth, ships should dwindle evenly until they disappear as a tiny featureless dot. That the masts and sails vanish first, Aristotle saw in a flash of genius, is a sign that the earth is curved. To observe the large-scale structure of our planet, Aristotle had looked through the window of geometry.

If something is true and you try to disprove it, you will fail. We are trained to to think of failure as bad, but it's not all bad. You can learn from failure. You try to disprove the statement one way, and you hit a wall. You try another way, and you hit another wall. Each night you try, each night you fail, each night a new wall, and if you are lucky, those walls start to come together into a structure, and that structure is the structure of the proof of the theorem. For if you have really understood what's keeping you from disproving the theorem, you very likely understand, in a way inaccessible to you before, why the theorem is true. This is what happened to Bolyai, who bucked his father's well-meaning advice and tried, like so many before him, to prove that the parallel postulate followed from Euclid's other axioms. Like all the others, he failed. But unlike the others, he was able to understand the shape of his failure. What was blocking all his attempts to prove that there was no geometry without the parallel postulate was the existence of just such a geometry! And with each failed attempt he learned more about the features of the thing he didn't think existed, getting to know it more and more intimately, until the moment when he realized it was really there.

It is hard to believe that the myths told about Pythagoras did not influence the creation of some of the later stories about Christ. Pythagoras, for instance, was believed by many to be the son of God, in this case, Apollo. His mother was called Parthenis, which means “virgin.” Before traveling to Egypt, Pythagoras lived the life of a hermit on Mount Carmel, like Christ's solitary vigil on the mountain. A Jewish sect, the Essenes, appropriated this myth and is said to have later had a connection to John the Baptist. There is also a myth that Pythagoras returned from the dead, although, according to the story, Pythagoras faked this by hiding in a secret underground chamber.

Por ejemplo, tomemos un segmento de 100 centímetros de largo y dividámoslo en dos partes de 61,8 centímetros y 38,2 centímetros. Con ello nos acercamos a la proporción áurea, porque 100 dividido entre 61,8 es casi lo mismo que 61,8 dividido entre 38,2; en ambos casos, más o menos, 1,618. Euclides escribió acerca de esta proporción hacia el año 300 a. C. y, desde entonces, fascina a los matemáticos. Pacioli fue el primero en popularizarlo con el nombre de «divina proporción». En su libro del mismo título, describió la forma en que aparece en los estudios de algunos cuerpos geométricos como cubos, prismas y poliedros. Según la creencia popular, incluido El código Da Vinci, de Dan Brown, la proporción áurea se halla en todo el arte de Leonardo.[11] Si es así, resulta, dudoso que lo hiciera de forma intencionada. Aunque pueden dibujarse esquemas de la Mona Lisa y de San Jerónimo que corroboran esta hipótesis, no existen pruebas irrefutables de que Leonardo utilizara conscientemente esta proporción matemática exacta. Sin embargo, el interés de Leonardo por las razones armónicas se refleja en sus exhaustivos estudios sobre el modo en que se manifiestan las proporciones en la anatomía, en la ciencia y en el arte, lo que le llevaría a buscar analogías entre las proporciones del cuerpo, las notas de las armonías musicales y otras proporciones que constituyen los cimientos sobre los que se apoya la belleza de las formas y los objetos de la naturaleza.

Q5. Have not I merely shown that it is possible to outdo just a particular algorithmic procedure, A, by defeating it with the computation Cq(n)? Why does this show that I can do better than any A whatsoever? The argument certainly does show that we can do better than any algorithm. This is the whole point of a reductio ad absurdum argument of this kind that I have used here. I think that an analogy might be helpful here. Some readers will know of Euclid's argument that there is no largest prime number. This, also, is a reductio ad absurdum. Euclid's argument is as follows. Suppose, on the contrary, that there is a largest prime; call it p. Now consider the product N of all the primes up to p and add 1: N=2*3*5*...*p+1. N is certainly larger than p, but it cannot be divisible by any of the prime numbers 2,3,5...,p (since it leaves the remainder 1 on division); so either N is the required prime itself or it is composite-in which case it is divisible by a prime larger than p. Either way, there would have to be a prime larger than p, which contradicts the initial assumption that p is the largest prime. Hence there is no largest prime. The argument, being a reductio ad absurdum, does not merely show that a particular prime p can be defeated by finding a larger one; it shows that there cannot be any largest prime at all. Likewise, the Godel-Turing argument above does not merely show that a particular algorithm A can be defeated, it shows that there cannot be any (knowably sound) algorithm at all that is equivalent to the insights that we use to ascertain that certain computations do not stop.

Many of the really great, famous proofs in the history of math have been reduction proofs. Here's an example. It is Euclid's proof of Proposition 20 in Book IX of the Elements. Prop. 20 concerns the primes, which-as you probably remember from school-are those integers that can't be divided into smaller integers w/o remainder. Prop. 20 basically states that there is no largest prime number. (What this means of course is that the number of prime numbers is really infinite, but Euclid dances all around this; he sure never says 'infinite'.) Here is the proof. Assume that there is in fact a largest prime number. Call this number Pn. This means that the sequence of primes (2,3,5,7,11,...,Pn) is exhaustive and finite: (2,3,5,7,11,...,Pn) is all the primes there are. Now think of the number R, which we're defining as the number you get when you multiply all the primes up to Pn together and then add 1. R is obviously bigger than Pn. But is R prime? If it is, we have an immediate contradiction, because we already assumed that Pn was the largest possible prime. But if R isn't prime, what can it be divided by? It obviously can't be divided by any of the primes in the sequence (2,3,5,...,Pn), because dividing R by any of these will leave the remainder 1. But this sequence is all the primes there are, and the primes are ultimately the only numbers that a non-prime can be divided by. So if R isn't prime, and if none of the primes (2,3,5,...,Pn) can divide it, there must be some other prime that divides R. But this contradicts the assumption that (2,3,5,...,Pn) is exhaustive of all the prime numbers. Either way, we have a clear contradiction. And since the assumption that there's a largest prime entails a contradiction, modus tollens dictates that the assumption is necessarily false, which by LEM means that the denial of the assumption is necessarily true, meaning there is no largest prime. Q.E.D.