| 单词 | conic section |
| 例句 | For centuries the Greeks had been fascinated with conic sections. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z They never worried about whether the diagonal of a square was rational or irrational, nor did they investigate the conic sections as Archimedes had. Zero: The Biography of a Dangerous Idea 2000-02-07T00:00:00Z But Newton was the first to propose an exact mathematical form and to use that form to show that the motion of heavenly bodies should be conic sections—circles, ellipses, parabolas, and hyperbolas. College Physics for AP Courses 2015-08-12T00:00:00Z This is why such curves are called conic sections. Astronomy 2016-10-13T00:00:00Z Objective 2: Identify conic sections by their equations. College Algebra with Corequisite Support 2020-09-23T00:00:00Z In this chapter we will be looking at the conic sections, usually called the conics, and their properties. Intermediate Algebra 2020-05-06T00:00:00Z When a quadric surface intersects a coordinate plane, the trace is a conic section. Calculus, Volume 3 2016-03-30T00:00:00Z The focal parameter of a conic section p is defined as the distance from a focus to the nearest directrix. Calculus, Vol 2 2016-03-30T00:00:00Z Write in the standard form of a conic if possible, and identify the conic section represented. Algebra and Trigonometry 2015-02-13T00:00:00Z It’s a somewhat sad development in a subject that began in Euclid’s day with the splendidly visual concepts of points, lines, triangles, circles, conic sections and the like. In Love with Geometry 2013-09-23T22:15:00.847Z Excluding the general group itself, every one of these leaves either a point, a line, or a conic section unaltered. Encyclopaedia Britannica, 11th Edition, Volume 12, Slice 6 "Groups, Theory of" to "Gwyniad" 2011-12-16T03:00:12.320Z We can view these surfaces as three-dimensional extensions of the conic sections we discussed earlier: the ellipse, the parabola, and the hyperbola. Calculus, Volume 3 2016-03-30T00:00:00Z Using the definitions of the focal parameter and eccentricity of the conic section, we can derive an equation for any conic section in polar coordinates. Calculus, Vol 2 2016-03-30T00:00:00Z The way in which we slice the cone will determine the type of conic section formed at the intersection. Algebra and Trigonometry 2015-02-13T00:00:00Z While these conic sections are not studied in elementary geometry, the terms should be known for general information, particularly the ellipse and parabola. The Teaching of Geometry 2011-10-12T02:00:52.133Z The comprehensive scheme of study included mathematics also, in which he advanced as far as the conic sections in the treatise of L’H�pital. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 8 "Germany" to "Gibson, William" 2011-10-05T02:00:17.763Z Quadric surfaces are three-dimensional surfaces with traces composed of conic sections. Calculus, Volume 3 2016-03-30T00:00:00Z In particular, we assume that one of the foci of a given conic section lies at the pole. Calculus, Vol 2 2016-03-30T00:00:00Z TRY IT #1 Identify the graph of each of the following nondegenerate conic sections. Algebra and Trigonometry 2015-02-13T00:00:00Z At this time, or in speaking of the preliminary definitions, reference should be made to the conic sections. The Teaching of Geometry 2011-10-12T02:00:52.133Z According to the commentator Proclus, the Elements were written with a twofold object, first, to introduce the novice to geometry, and secondly, to lead him to the regular solids; conic sections found no place therein. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z When I left Woolwich my coarse in mathematics consisted of plane and spherical trigonometry, conic sections, statics and dynamics, properties of roofs and arches, hydrostatics, projectiles, and the deferential and integral calculus. The Gentleman Cadet His Career and Adventures at the Royal Military Academy Woolwich 2011-08-31T02:01:40.423Z Then using the definition of the various conic sections in terms of distances, it is possible to prove the following theorem. Calculus, Vol 2 2016-03-30T00:00:00Z Each of these orbits can be modeled by a conic section in the polar coordinate system. Algebra and Trigonometry 2015-02-13T00:00:00Z The study of conic sections was brought to a high state by the Greeks. The Teaching of Geometry 2011-10-12T02:00:52.133Z What Euclid did for the line and circle, Apollonius did for the conic sections, but there we have a discoverer as well as editor. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z I was beginning conic sections in the third half-year, and this subject I found was one that I could manage very well by thinking quietly over. The Gentleman Cadet His Career and Adventures at the Royal Military Academy Woolwich 2011-08-31T02:01:40.423Z In the equation on the left, the major axis of the conic section is horizontal, and in the equation on the right, the major axis is vertical. Calculus, Vol 2 2016-03-30T00:00:00Z We must use the eccentricity of a conic section to determine which type of curve to graph, and then determine its specific characteristics. Algebra and Trigonometry 2015-02-13T00:00:00Z It only remains to add that Archimedes is fully acquainted with the main properties of the conic sections. Archimedes Men of Science 2011-03-13T03:00:23.660Z The conic sections are the next simplest loci; and it will be seen later that they are the loci represented by equations of the second degree. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z Johnson studied Euclid, algebra, and conic sections "so as to read Sir Isaac with understanding." Benjamin Franklin Representative selections, with introduction, bibliograpy, and notes 2011-03-08T03:00:46.777Z The graph of an equation of this form is a conic section. Calculus, Vol 2 2016-03-30T00:00:00Z Explain how eccentricity determines which conic section is given. Algebra and Trigonometry 2015-02-13T00:00:00Z It would appear that it was in the effort to solve this problem that Men�chmus discovered the conic sections, which are called, in an epigram by Eratosthenes, “the triads of Men�chmus”. Archimedes Men of Science 2011-03-13T03:00:23.660Z Circles are particular cases of conic sections; and 714 they have just been seen to have for their equations a particular class of equations of the second degree. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 "Geodesy" to "Geometry" 2011-09-19T02:00:10.473Z The relation of the ellipse to the other conic sections is treated in the articles Conic Section and Geometry; in this article a summary of the properties of the curve will be given. Encyclopaedia Britannica, 11th Edition, Volume 9, Slice 3 "Electrostatics" to "Engis" 2011-02-06T03:00:53.093Z The method for graphing a conic section with rotated axes involves determining the coefficients of the conic in the rotated coordinate system. Calculus, Vol 2 2016-03-30T00:00:00Z If a conic section is written as a polar equation, what must be true of the denominator? Algebra and Trigonometry 2015-02-13T00:00:00Z He developed especially what we know now as conic sections. Education: How Old The New 2011-01-14T03:00:54.370Z It is the same shape that you would find if you were to cut a cone through with a saw, and so these curves which I have shown you are called conic sections. Soap-Bubbles and the Forces Which Mould Them At ten minutes to three Trinkmann stood behind the cashier's desk, so thoroughly enmeshed in the intricacies of his wife's bookkeeping that not even a knowledge of conic sections would have disentangled him. The Competitive Nephew Appollonius wrote an entire eight-volume treatise on conic sections in which he was, for example, able to derive a specific method for identifying a conic section through the use of geometry. Calculus, Vol 2 2016-03-30T00:00:00Z For the following exercises, convert the polar equation of a conic section to a rectangular equation. Algebra and Trigonometry 2015-02-13T00:00:00Z His book on the subject contains practically all of the theorems to be found in our text-books of analytical geometry or conic sections of the present time. Education: How Old The New 2011-01-14T03:00:54.370Z We are now almost able to see what the conic section has to do with a soap-bubble. Soap-Bubbles and the Forces Which Mould Them Pascal, at twelve, had completely mastered Euclid's Elements without any assistance, and at sixteen he published a work on conic sections, which Descartes was reluctant to believe had been produced by a boy. The Scrap Book, Volume 1, No. 5 July 1906 In this section we discuss the three basic conic sections, some of their properties, and their equations. Calculus, Vol 2 2016-03-30T00:00:00Z For the following exercises, determine which conic section is represented by the given equation, and then determine the angle that will eliminate the term. Algebra and Trigonometry 2015-02-13T00:00:00Z Equations, conic sections, differential calculus, constitute a skull and cross-bones to which I allow as wide a berth as possible.” Vashti or, Until Death Us Do Part To propose philosophy to them, would be just as weak as to propose the study of conic sections to peasants or fish-women. Ancient and Modern Celebrated Freethinkers Reprinted From an English Work, Entitled "Half-Hours With The Freethinkers." Many of them correspond to conic sections, but it seems probable that the outlines were drawn by hand, and not obtained by any mechanical or mathematical method. Architecture Classic and Early Christian If the plane is parallel to the generating line, the conic section is a parabola. Calculus, Vol 2 2016-03-30T00:00:00Z As we have seen, conic sections are formed when a plane intersects two right circular cones aligned tip to tip and extending infinitely far in opposite directions, which we also call a cone. Algebra and Trigonometry 2015-02-13T00:00:00Z The fact that the force varies as the inverse square of the distance, necessitates motion in an ellipse, or some other conic section, with the sun in one focus. Pioneers of Science I would advise you to resort at once to geometry or conic sections, or some other equally inexorable discipline to settle the business. The Atlantic Monthly, Volume 20, No. 121, November, 1867 By this time I had studied plane and spherical trigonometry, conic sections, and Fergusson's "Astronomy." Personal Recollections, from Early Life to Old Age, of Mary Somerville If the plane is perpendicular to the axis of revolution, the conic section is a circle. Calculus, Vol 2 2016-03-30T00:00:00Z Depending on the angle of the plane, three types of degenerate conic sections are possible: a point, a line, or two intersecting lines. Algebra and Trigonometry 2015-02-13T00:00:00Z The curve which represents with perfect fidelity the movements of a planet in its revolution around the sun belongs to that well-known group of curves which mathematicians describe as the conic sections. The Story of the Heavens He proposed to design linkages to describe a straight line, a circle of any radius no matter how large, and conic sections, and he indicated in his letter that he had arrived at a solution. Kinematics of Mechanisms from the Time of Watt In order to do this, we must first determine the social properties of a conic section. The Romance of Mathematics Being the Original Researches of a Lady Professor of Girtham College in Polemical Science, with some Account of the Social Properties of a Conic; Equations to Brain Waves; Social Forces; and the Laws of Political Motion. The three conic sections with their directrices appear in the following figure. Calculus, Vol 2 2016-03-30T00:00:00Z Equations of conic sections with an term have been rotated about the origin. Algebra and Trigonometry 2015-02-13T00:00:00Z The particular form of conic section which denotes the orbit of a planet is known by the name of the ellipse: it is spoken of somewhat less accurately as an oval. The Story of the Heavens It is a property of conic sections that a straight line, centred in the apex, and caused to circumscribe the surface of the cone, will apply itself continuously to all consecutive parabolic curves. The Life of Thomas, Lord Cochrane, Tenth Earl of Dundonald, Vol. II The simplest conic section of all has been proved to be a point. The Romance of Mathematics Being the Original Researches of a Lady Professor of Girtham College in Polemical Science, with some Account of the Social Properties of a Conic; Equations to Brain Waves; Social Forces; and the Laws of Political Motion. To work with a conic section written in polar form, first make the constant term in the denominator equal to 1. Calculus, Vol 2 2016-03-30T00:00:00Z Because the discriminant is invariant, observing it enables us to identify the conic section. Algebra and Trigonometry 2015-02-13T00:00:00Z This curve is not answerable to any of the figures of conic sections, although it occasionally partakes of them all. The Sailor's Word-Book An Alphabetical Digest of Nautical Terms, including Some More Especially Military and Scientific, but Useful to Seamen; as well as Archaisms of Early Voyagers, etc. Though but sixteen, the youth had already attracted the notice of the mathematical world by his treatise on conic sections. The Story of Rouen So he hastily worked the sum backward in decimals and equations and conic sections, and all the rules he could think of. The Book of Dragons She can prove her statement in conic sections or algebra, but when it comes to actual knowledge she is deficient. As A Chinaman Saw Us Passages from his Letters to a Friend at Home For the following exercises, graph the given conic section. Algebra and Trigonometry 2015-02-13T00:00:00Z Only the extremely ignorant can question the utility of, let us say, the prolonged application of the Greek intellect to the laws of conic sections. Human Traits and their Social Significance Menaechmus, a pupil of Eudoxus, was the discoverer of the conic sections, two of which, the parabola and the hyperbola, he used for solving the problem of the two mean proportionals. The Legacy of Greece Essays By: Gilbert Murray, W. R. Inge, J. Burnet, Sir T. L. Heath, D'arcy W. Thompson, Charles Singer, R. W. Livingston, A. Toynbee, A. E. Zimmern, Percy Gardner, Sir Reginald Blomfield To have expected the old nurse to take an enlightened view on that point would have been as unreasonable as to have looked for just views in Gillie White on the subject of conic sections. Rivers of Ice They might be a treatise on conic sections, or a Latin Grammar, and again they might be a Book. The Bibliotaph and Other People Later in this chapter, we will see that the graph of any quadratic equation in two variables is a conic section. Algebra and Trigonometry 2015-02-13T00:00:00Z Whether we think of bridges or projectiles, of the curves of ships, or of the rules of navigation, we must think of conic sections. Human Traits and their Social Significance Yet it may return—apparently contradicting the geometry of conic sections. Spacehounds of IPC Parabola.—One of the conic sections formed by cutting of a cone so that the cut line, A, is not vertical. Carpentry for Boys In a Simple Language, Including Chapters on Drawing, Laying Out Work, Designing and Architecture With 250 Original Illustrations Blaise Pascal, the French mathematician, composed at sixteen a tract on the conic sections. How to Succeed or, Stepping-Stones to Fame and Fortune Because the discriminant remains unchanged, observing the discriminant enables us to identify the conic section. Algebra and Trigonometry 2015-02-13T00:00:00Z Kepler's Laws are foundation stones of that science, but Kepler discovered that Mars moves in an ellipse round the sun in one of the foci by a deduction from conic sections.... Human Traits and their Social Significance One might as reasonably go to the Louvre for classes in conic sections, or to the Garden of the Gods for instruction in Rabbinical theology. The Joyful Heart It would be suggesting to fishwives that they should study conic sections. Voltaire's Philosophical Dictionary Eggs à la reine seemed as difficult as trigonometry, or conic sections, or differential calculus—and much more expensive. The Wit and Humor of America, Volume I. (of X.) What effect does the term have on the graph of a conic section? Algebra and Trigonometry 2015-02-13T00:00:00Z The only sure cases are mathematical; as we may show that there are possible only three kinds of plane triangles, four conic sections, five regular solids. Logic Deductive and Inductive To him planetary and selective transmission were even as conic sections. Potash & Perlmutter Their Copartnership Ventures and Adventures We proceed to develop some theorems which will exhibit the importance of these points in the theory of the conic section. An Elementary Course in Synthetic Projective Geometry I sat down to warm my hands and my half-frozen face, and when I felt comfortable I said, "Do you read conic sections, young gentleman?" The Romance of the Coast For the following exercises, determine which conic section is represented based on the given equation. Algebra and Trigonometry 2015-02-13T00:00:00Z Would it not be very hard to expect anyone to solve a problem in conic sections who had merely been taught the axioms and definitions of mathematical science? Lay Sermons, Addresses and Reviews It is far more pleasant to be describing a shield and settling questions in the queer old language of this queer old science, than in solving and propounding problems in trigonometry and conic sections. In Luck at Last We find in this little book the beautiful theorem concerning a quadrilateral inscribed in a conic section, which is given by his name in § 138. An Elementary Course in Synthetic Projective Geometry She first put me into the ABC class, and advanced me through conic sections. Among the Forces If the directrix of a conic section is perpendicular to the polar axis, what do we know about the equation of the graph? Algebra and Trigonometry 2015-02-13T00:00:00Z Shooting stars, fire-balls, and meteoric stones are, with great probability, regarded as small bodies moving with planetary velocity, and revolving in obedience to the laws of general gravity in conic sections round the Sun. COSMOS: A Sketch of the Physical Description of the Universe, Vol. 1 Pascal showed the most astonishing mathematical genius; he produced at the age of seventeen a profound work on conic sections, and devoted the following years to physical researches and to investigations in the higher mathematics. The World's Greatest Books — Volume 13 — Religion and Philosophy Cut across, now, by any plane, and we get a conic section which is thus exhibited as the locus of intersection of two projective pencils. An Elementary Course in Synthetic Projective Geometry It appears, then, that a conic section may be defined as the locus of a point which is equally distant from a given point and from the circumference of a given circle. Scientific American Supplement, No. 803, May 23, 1891 What do we know about the focus/foci of a conic section if it is written as a polar equation? Algebra and Trigonometry 2015-02-13T00:00:00Z "My friend," answered the captain, "a parabola is a conic section arising from cutting a cone by a plane parallel to one of its sides." The Moon-Voyage "I think I'm beginning to see real light in conic sections, but I'll have the books out again tomorrow afternoon." Dick Prescott's Second Year at West Point Finding the Glory of the Soldier's Life It will later appear that a point-row of the second order is a conic section. An Elementary Course in Synthetic Projective Geometry MACCORD.—Examination of the four conic sections with a general definition applicable to all.—6 illustrations VII. Scientific American Supplement, No. 803, May 23, 1891 For the following exercises, graph the given conic section. Algebra and Trigonometry 2015-02-13T00:00:00Z Parabola, a conic section formed by the intersection of a cone by a plane parallel to one of its sides. The Nuttall Encyclopædia Being a Concise and Comprehensive Dictionary of General Knowledge You'll bone yourself dry, staying here all the time with Smith's conic sections. Dick Prescott's Second Year at West Point Finding the Glory of the Soldier's Life The converse is also true, as we shall see later, and a pencil of rays of the second order is also a set of lines tangent to a conic section. An Elementary Course in Synthetic Projective Geometry Undoubtedly, in the straight line, in the conic sections, in the innumerable composite curves of the mathematician, lie the germs of all these symbolic expressions. The Atlantic Monthly, Volume 07, No. 44, June, 1861 Creator Besides theological writings, he contributed much to mathematical science, especially in the directions of conic sections, analytic geometry, higher plane curves, and the geometry of three dimensions. The Glories of Ireland It was a delightful task for him to lead them through the pleasant valleys of conic sections, and beside the still waters of the integral calculus. Stories by American Authors, Volume 5 Parabolic Curves.—Having the form of a parabola—a conic section. Flying Machines: construction and operation; a practical book which shows, in illustrations, working plans and text, how to build and navigate the modern airship It was not till 1522 that Verner3 of Nürnberg undertook to demonstrate the properties of the conic sections by means of the circle. An Elementary Course in Synthetic Projective Geometry He found it impossible to study conic sections; something in their calm and tantalizing respectability breathing defiantly through Mr. Rooney's fetid parlors distorted their equations into insoluble anagrams. This Side of Paradise The conic sections are the parabola, ellipse, and hyperbola. The Gutenberg Webster's Unabridged Dictionary Section C And if we seek a law for the ellipse, or either of the conic sections, which shall confine our attention to the plane, the laws remain simple. The Atlantic Monthly, Volume 05, No. 30, April, 1860 Pascal wrote treatises on conic sections at sixteen and invented his arithmetical machine at nineteen. Youth: Its Education, Regimen, and Hygiene It thus appears that a conic section is a point-row of the second order. An Elementary Course in Synthetic Projective Geometry It seemed a stupid way to commence his upper-class years, to spend four hours a morning in the stuffy room of a tutoring school, imbibing the infinite boredom of conic sections. This Side of Paradise At the age of sixteen he wrote a treatise on conic sections, which was followed by several important discoveries in arithmetic and geometry. Handbook of Universal Literature From the Best and Latest Authorities The study of the conic sections has been a part of polite learning from his day downward. The Atlantic Monthly, Volume 05, No. 30, April, 1860 Spherical trigonometry and conic sections are Arabic inventions. Early European History By projecting the figure thus described upon another plane he is able to state his theorem for the case where the circle is replaced by any conic section. An Elementary Course in Synthetic Projective Geometry Others were rejected on the fifth or sixth day as being deficient in conic sections, or ignorant of the exact principles of hydraulic pressure. The Three Clerks Would it not be very hard to expect any one to solve a problem in conic sections who had merely been taught the axioms and definitions of mathematical science? Science & Education His most important work is a treatise on conic sections, which was regarded with unbounded admiration by contemporaries, and in some respects is unsurpassed by any thing produced by modern mathematicians. Beacon Lights of History, Volume 03 Ancient Achievements The disciples of Plato invented conic sections, and discovered the geometrical loci. The Old Roman World, : the Grandeur and Failure of Its Civilization. All conic sections, e.g. ellipses, parabolas, and hyperbolas, could be represented by equations. Our Legal Heritage : 600-1776 King Aethelbert - King George III It may one day be conic sections, another Greek iambics, and a third German philosophy. The Three Clerks How they stared at me, Those men of "sensibility," when I said That algebra, conic sections, fluxions, all Pertained to music. Watchers of the Sky The disciples of Plato invented conic sections, and discovered the geometrical foci. Beacon Lights of History, Volume 03 Ancient Achievements His most important work is a treatise on conic sections, regarded with unbounded admiration by contemporaries, and, in some respects, unsurpassed by any thing produced by modern mathematicians. The Old Roman World, : the Grandeur and Failure of Its Civilization. She thought at first he was dead, for there came no answer when she spoke; but presently she heard him murmur something about conic sections. Warlock o' Glenwarlock It was very clear that the young gentleman who had only showed that he had forgotten his conic sections could not be supposed to have passed. The Three Clerks He did not disbelieve it; he would have been shocked to hear such a thing asserted of him: but he happened to be busy believing something else—geometry, conic sections, cosmogonies, psychologies, and what not. Hypatia — or New Foes with an Old Face Culture came to him first, while yet he abode in Philistia, under the playful disguise of a conic section. Philistia I had never learned the multiplication table, and had no more idea of the rule of three than of conic sections. Autobiography of Anthony Trollope A few minutes more and the two were again absorbed, this time with a point in conic sections, on which Aggie professed to require enlightenment, and again Elsie was left out. Warlock o' Glenwarlock So we may say that there are different planes of Memory, conic sections, if we keep up the original metaphor, and the largest of these contains all our past. Bergson and His Philosophy He also thought in conic sections, squares and roots and ratios, and geometrized like Euclid. Pragmatism In its composition there may be fifty, more or less cohering, conic sections, each containing an unborn shell in a distinct and separate stage of development. Confessions of a Beachcomber He lay there wide awake, with disheveled hair and an inspired gleam in his eye, tracing on the floor, with the point of a corkscrew, what looked like a tangle of parallelograms and conic sections. South Wind If he is not satisfied with that I refer him to any work on conic sections and if not convinced even then I refer him so far that he will never come back. My Discovery of England Utterly worn out by his course of conic sections, my chum declares that he has had enough. The Life of the fly; with which are interspersed some chapters of autobiography Greek accents, however, and conic sections were esteemed necessaries at Balliol, and there was no admittance there for Mr. Arabin within the list of its fellows. Barchester Towers For indeed these superficies of conic sections are circles. Complete Works of Plutarch — Volume 3: Essays and Miscellanies I had seamanship, gunnery, navigation, nautical astronomy, algebra, geometry, trigonometry, conic sections, curve tracing, differential and integral calculus. Real Soldiers of Fortune Working hard, too, helped him not a little, and conic sections served him almost as well as they served Laura. The Heir of Redclyffe The conic sections which I described on that blackboard, the learned hieroglyphics! The Life of the fly; with which are interspersed some chapters of autobiography Greek accents, however, and conic sections were esteemed necessaries at Balliol, and there was no admittance there for Mr Arabin within the list of its fellows. Barchester Towers When he was sixteen we learn that he had read conic sections, and that he was engaged in the study of pendulums. Great Astronomers I had a natural distaste for mathematics, and my recollections of my struggles with trigonometry and conic sections are not altogether those of a conquering heroine. A New England girlhood, outlined from memory (Beverly, MA) Numerous other of the studies of Archimedes having reference to conic sections, properties of curves and spirals, and the like, are too technical to be detailed here. A History of Science — Volume 1 |
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