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...
Let me tell you how at one time the famous mathematician Euclid became a physician. It was during a vacation, which I spent in Prague as I most always did, when I was attacked by an illness never before experienced, which manifested itself in chilliness and painful weariness of the whole body. In order to ease my condition I took up Euclid's Elements and read for the first time his doctrine of ratio, which I found treated there in a manner entirely new to me. The ingenuity displayed in Euclid's presentation filled me with such vivid pleasure, that forthwith I felt as well as ever.
I claim that many patterns of Nature are so irregular and fragmented, that, compared with Euclid-a term used in this work to denote all of standard geometry-Nature exhibits not simply a higher degree but an altogether different level of complexity ... The existence of these patterns challenges us to study these forms that Euclid leaves aside as being "formless," to investigate the morphology of the "amorphous."
Euclid alone has looked on Beauty bare. Let all who prate of Beauty hold their peace, And lay them prone upon the earth and cease To ponder on themselves, the while they stare At nothing, intricately drawn nowhere In shapes of shifting lineage; let geese Gabble and hiss, but heroes seek release From dusty bondage into luminous air. O blinding hour, O holy, terrible day, When first the shaft into his vision shone Of light anatomized! Euclid alone Has looked on Beauty bare. Fortunate they Who, though once only and then but far away, Have heard her massive sandal set on stone.
Edna St. Vincent Millay
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 Je¡nos from any further attempt.]
The existence of these patterns [fractals] challenges us to study forms that Euclid leaves aside as being formless, to investigate the morphology of the amorphous. Mathematicians have disdained this challenge, however, and have increasingly chosen to flee from nature by devising theories unrelated to anything we can see or feel.
It would be foolish to give credit to Euclid for pangeometrical conceptions; the idea of geometry deifferent from the common-sense one never occurred to his mind. Yet, when he stated the fifth postulate, he stood at the parting of the ways. His subconscious prescience is astounding. There is nothing comperable to it in the whole history of science.
Like a young heir, come a little prematurely to a large inheritance, we shall wanton and run riot until we have brought our reputation to the brink of ruin, and then, like him, shall have to labor with the current of opinion, when COMPELLED perhaps, to do what prudence and common policy pointed out, as plain as any problem in Euclid, in the first instance.
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.
At the age of eleven, I began Euclid, with my brother as my tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world. From that moment until I was thirty-eight, mathematics was my chief interest and my chief source of happiness.
My venture investing career has three phases, all roughly 6-8 years long. The first, at Euclid, was software to Internet. The second, at Flatiron, was Internet to bubble. And the third, at USV, has been web 2 to mobile. I have always used a new firm to denote a new investment phase for me. Throw away the old. Start with the new.
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 Je¡nos from any further attempt.]
We think of Euclid as of fine ice; we admire Newton as we admire the peak of Teneriffe. Even the intensest labors, the most remote triumphs of the abstract intellect, seem to carry us into a region different from our own-to be in a terra incognita of pure reasoning, to cast a chill on human glory.
It is curious to observe the triumph of slight incidents over the mind; and what incredible weight they have in forming and governing our opinions, both of men and things, that trifles light as air shall waft a belief into the soul, and plant it so immovable within it, that Euclid's demonstrations, could they be brought to batter it in breach, should not all have power to overthrow it!
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.
I told myself, "Lincoln, you can never make a lawyer if you do not understand what demonstrate means." So I left my situation in Springfield, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what "demonstrate" means, and went back to my law studies.
The new painters do not propose, any more than did their predecessors, to be geometers. But it may be said that geometry is to the plastic arts what grammar is to the art of the writer. Today, scholars no longer limit themselves to the three dimensions of Euclid. The painters have been lead quite naturally, one might say by intuition, to preoccupy themselves with the new possibilities of spatial measurement which, in the language of the modern studios, are designated by the term fourth dimension.
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.
The science of the church is neglected for the study of geometry, and they lose sight of Heaven while they are employed in measuring the earth. Euclid is perpetually in their hands. Aristotle and Theophrastus are the objects of their admiration; and they express an uncommon reverence for the works of Galen. Their errors are derived from the abuse of the arts and sciences of the infidels, and they corrupt the simplicity of the gospel by the refinements of human reason.
In geometry I find certain imperfections which I hold to be the reason why this science, apart from transition into analytics, can as yet make no advance from that state in which it came to us from Euclid.As belonging to these imperfections, I consider the obscurity in the fundamental concepts of the geometrical magnitudes and in the manner and method of representing the measuring of these magnitudes, and finally the momentous gap in the theory of parallels, to fill which all efforts of mathematicians have so far been in vain.
Blaise Pascal used to mark with charcoal the walls of his playroom, seeking a means of making a circle perfectly round and a triangle whose sides and angle were all equal. He discovered these things for himself and then began to seek the relationship which existed between them. He did not know any mathematical terms and so he made up his own. Using these names he made axioms and finally developed perfect demonstrations, until he had come to the thirty-second proposition of Euclid.
Catharine Cox Miles
It seems that mathematical ideas are arranged somehow in strata, the ideas in each stratum being linked by a complex of relations both among themselves and with those above and below. The lower the stratum, the deeper (and in general more difficult) the idea. Thus the idea of an 'irrational' is deeper than that of an integer; and Pythagoras's theorem is, for that reason, deeper than Euclid's.
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.
Mathematics has two faces: it is the rigorous science of Euclid, but it is also something else. Mathematics presented in the Euclidean way appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science. Both aspects are as old as the science of mathematics itself.
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 and magicians; instead, at the origin of physics lay the strenuous clarity of the West-Archimedes and Euclid.
When one day Lagrange took out of his pocket a paper which he read at the Academe, and which contained a demonstration of the famous Postulatum of Euclid, relative to the theory of parallels. This demonstration rested on an obvious paralogism, which appeared as such to everybody; and probably Lagrange also recognised it such during his lecture. For, when he had finished, he put the paper back in his pocket, and spoke no more of it. A moment of universal silence followed, and one passed immediately to other concerns.
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.
About Thomas Hobbes: He was 40 years old before he looked on geometry; which happened accidentally. Being in a gentleman's library, Euclid's Elements lay open, and "twas the 47 El. libri I" [Pythagoras' Theorem]. He read the proposition "By God", sayd he, "this is impossible:" So he reads the demonstration of it, which referred him back to such a proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps, that at last he was demonstratively convinced of that truth. This made him in love with geometry.
The anceints devoted a lifetime to the study of arithmetic; it required days to extract a square root or to multiply two numbers together. Is there any harm in skipping all that, in letting the school boy learn multiplication sums, and in starting his more abstract reasoning at a more advanced point. Where would be the harm in letting the boy assume the truth of many propositions of the first four books of Euclid, letting him assume their truth partly by faith, partly by trial?
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.
These estimates may well be enhanced by one from F. Klein (1849-1925), the leading German mathematician of the last quarter of the nineteenth century. 'Mathematics in general is fundamentally the science of self-evident things.'... If mathematics is indeed the science of self-evident things, mathematicians are a phenomenally stupid lot to waste the tons of good paper they do in proving the fact. Mathematics is abstract and it is hard, and any assertion that it is simple is true only in a severely technical sense-that of the modern postulational method which, as a matter of fact, was exploited by Euclid. The assumptions from which mathematics starts are simple; the rest is not.
Eric Temple Bell
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.
William Kingdon Clifford
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.
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.
Charles Sanders Peirce
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.
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?
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 e†schylus 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?
Robert G. Ingersoll
Reading list (1972 edition) 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 Nje¡l 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. Niccole² 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. Frane§ois 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. Rene Descartes - Rules for the Direction of the Mind; Discourse on the Method; Geometry; Meditations on First Philosophy 58. John Milton - Works 59. Molie¨re - 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
Mortimer J. Adler