The Ellipse: A Historical and Mathematical Journey
For an ellipse of the same eccentricity as the Earth Meridian, such a relative error is only about 4. For an ellipse the size of the Earth's Meridian, this would mean a ludicrous precision of 1. Ellipses of low eccentricities are thus well taken care of, by either of the above approximations Ramanujan's formula is always much better.
For very elongated ellipses, Ramanujan's formula is thus about 10 times better than our first approximation and may be somewhat acceptable in practice, although the error figure remains unimpressive.
Lindner's formula estimates the perimeter of a flat ellipse with a relative error of about Following our established pattern, we may expand this formula as a power series of e 2. This is a religious experience! All the terms match the correct series up to and including the coefficient of e It takes some bravery to work out a precise expression for the relative error involved. Here it is:. That's way beyond comparison with any available physical yardstick: The physical Universe the one we live in is thought to lack any smoothness at the Planck Scale , where dimensions are on the order of the Planck Length , a unit roughly equal to 1.
However, that may not be much in view of the fact that ordinary geometry surely breaks down in the physical Universe well before the Planck realm is reached Ralph G. It yields a relative error of about This formula appears without attribution on page 17 of the sixth printing of " A Manual of Mathematics " by Ralph G. Cantrell for finding this out.
That truncated sum agrees with the aforementioned "Hudson formula" but a discrepancy would occur with the very next term. This is directly comparable with the accuracy of Ramanujan's second formula 3 , which is 5. Hudson's formula 4 is one of only two circumference formulas quoted in the Handbook of Bronshtein and Semendyayev. Other circumference formulas have been proposed to minimize the relative error on some extended range of eccentricities.
D may also be optimized according to various other criteria. In particular, the value of D which yields the lowest maximal relative error over the entire range of eccentricities is slightly less than 2. For a nearly round ellipse, the precision of this formula is virtually the same as Euler's except that the error is negative instead of positive. Cantrell proposed another type of formula, which may be optimized like our quadratic formula 6 , but with numerical results that turn out to be more than 50 times better!
See , , etc. The value of p may thus be optimized for different criteria. Cantrell observes that an exponent p of about 0. We summarize the precisions of many of these approximations in the table below, where the first entry is the simple formula given by Johannes Kepler in as a lower bound to the perimeter of an ellipse, according to Almkvist and Berndt :.
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Fontenelle, Bernard Le Bovier A work of morality, politics, criticism will be more elegant, other things being equal, if it is shaped by the hand of geometry. Fontenelle, Bernard Le Bovier Leibniz never married; he had considered it at the age of fifty; but the person he had in mind asked for time to reflect.
This gave Leibniz time to reflect, too, and so he never married.
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Elog e de le Leibniz. Frankland, W. Whereas at the outset geometry is reported to have concerned herself with the measurement of muddy land, she now handles celestial as well as terrestrial problems: she has extended her domain to the furthest bounds of space. Hodder and Stoughton, The Story of Euclid. Frayn, Michael For hundreds of pages the closely-reasoned arguments unroll, axioms and theorems interlock.
And what remains with us in the end? A general sense that the world can be expressed in closely reasoned arguments, in interlocking a xioms and theorems. Frederick the Great To your care and recommendation am I indebted for having replaced a half-blind mathematician with a mathematician with both eyes, which will especially please the anatomical members of my Academy.
Euler had vacated the post. Frege, Gottlob - A scientist can hardly meet with anything more undesirable than to have the foundations give way just as the work is finished. I was put in this position by a letter from Mr.go to site
Bertrand Russell when the work was n early through the press. In Scientific American , May , p Galbraith, John Kenneth There can be no question, however, that prolonged commitment to mathematical exercises in economics can be damaging. It leads to the atrophy of judgement and intuition Economics, Peace, and Laughter. It is written in mathematical language, and the letters are triangles, circles an d other geometrical figures, without which means it is humanly impossible to comprehend a single word.
Opere Il Saggiatore p. Galilei, Galileo - Measure what is measurable, and make measurable what is not so.
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Quoted in H. Galilei, Galileo - And who can doubt that it will lead to the worst disorders when minds created free by God are compelled to submit slavishly to an outside will? When we are told to deny our senses and subject them to the whim of others? When people devoid of whatsoever competence are made judges over experts and are granted authority to treat them as they please?
These are the novelties which are apt to bring about the ruin of commonwealths and the subversion of the state. Galois, Evariste Unfortunately what is little recognized is that the most worthwhile scientific books are those in which the author clearly indicates what he does not know; for an author most hurts his readers by concealing difficulties.
Galton, [Sir] Francis Whenever you can, count.
The Ellipse: A Historical And Mathematical Journey
Galton, Sir Francis [Statistics are] the only tools by which an opening can be cut through the formidable thicket of difficulties that bars the path of those who pursue the Science of Man. Galton, Sir Francis I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the "Law of Frequency of Error. It reigns with serenity and in complete self effacement, amidst the wildest confusion.
The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshaled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along.
Gardner, Martin Biographical history, as taught in our public schools, is still largely a history of boneheads: ridiculous kings and queens, paranoid political leaders, compulsive voyagers, ignorant generals -- the flotsam and jetsam of his torical currents. The men who radically altered history, the great scientists and mathematicians, are seldom mentioned, if at all. Gardner, Martin Mathematics is not only real, but it is the only reality.
That is that entire universe is made of matter, obviously. And matter is made of particles. It's made of electrons and neutrons and protons. So the entire universe is made out of particles.
- The Ellipse. A historical and mathematical journey.
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Now what are the particles made out of? They're not made out of anything. The only thing you can say about the reality of an electron is to cite its mathematical properties. So there's a sense in which matter has completely dissolv ed and what is left is just a mathematical structure. Gauss, Karl Friedrich I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.
Gauss, Karl Friedrich If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries. Gauss, Karl Friedrich There are problems to whose solution I would attach an infinitely greater importance than to those of mathematics, for example touching ethics, or our relation to God, or concerning our destiny and our futu re; but their solution lies wholly beyond us and completely outside the province of science. Gauss, Karl Friedrich You know that I write slowly. This is chiefly because I am never satisfied until I have said as much as possible in a few words, and writing briefly takes far more time than writing at length.
Gauss, Karl Friedrich We must admit with humility that, while number is purely a product of our minds, space has a reality outside our minds, so that we cannot completely prescribe its properties a priori. Letter to Bessel, 1 Gauss, Karl Friedrich I have had my results for a long time: but I do not yet know how I am to arrive at them. Arber The Mind and the Eye Gauss, Karl Friedrich [His second motto:] Thou, nature, art my goddess; to thy laws my services are bound Shakespeare King Lear.
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Gauss, Karl Friedrich [attributed to him by H. Foreword of H. Gauss, Karl Friedrich It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in or der to go into darkness again; the never-satisfied man is so strange if he has completed a structure, then it is not in order to dwell in it peacefully, but in order to begin another.
I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretches out his arms for others.