# Conic Sentence Examples

John Casey, professor of mathematics at the Catholic university of Dublin, has given elementary demonstrations founded on the theory of similitude and coaxal circles which are reproduced in his Sequel to Euclid; an analytical solution by Gergonne is given in Salmon's

**Conic**Sections.It is therefore a

**conic**section having its eccentricity equal to unity.Cubic equations were solved geometrically by determining the intersections of

**conic**sections.In projective geometry it may be defined as the

**conic**which intersects the line at infinity in two real points, or to which it is possible to draw two real tangents from the centre.The same name is also given to the first positive pedal of any central

**conic**. When the**conic**is a rectangular hyperbola, the curve is the lemniscate of Bernoulli previously described.If the attraction of a central body is not the only force acting on the moving body, the orbit will deviate from the form of a

**conic**section in a degree depending on the amount of the extraneous force; and the curve described may not be a re-entering curve at all, but one winding around so as to form an indefinite succession of spires.The northern part of Eure-et-Loir is watered by the Eure, with its tributaries the Vegre, Blaise and Avre, a small western portion by the Huisne, and the south by the Loir with its tributaries the

**Conic**and the Ozanne.The Arabians more closely resembled the Hindus than the Greeks in the choice of studies; their philosophers blended speculative dissertations with the more progressive study of medicine; their mathematicians neglected the subtleties of the

**conic**sections and Diophantine analysis, and applied themselves more particularly to perfect the system of numerals, arithmetic and astronomy.He improved the methods for solving equations, and devised geometrical constructions with the aid of the

**conic**sections.The founder of the mathematical school was the celebrated Euclid (Eucleides); among its scholars were Archimedes; Apollonius of Perga, author of a treatise on

**Conic**Sections; Eratosthenes, to whom we owe the first measurement of the earth; and Hipparchus, the founder of the epicyclical theory of the heavens, afterwards called the Ptolemaic system, from its most famous expositor, Claudius Ptolemaeus.AdvertisementArchytas of Tarentum (c. 430 B.C.) solved the problems by means of sections of a half cylinder; according to Eutocius, Menaechmus solved them by means of the intersections of

**conic**sections; and Eudoxus also gave a solution.It is the inverse of a central

**conic**for the focus, and the first positive pedal of a circle for any point.The line la+ma+ny is the radical axis, and since as+43 c-y =o is the line infinity, it is obvious that equation (I) represents a

**conic**passing through the circular points, i.e.This problem, also termed the " Apollonian problem," was demonstrated with the aid of

**conic**sections by Apollonius in his book on Contacts or Tangencies; geometrical solutions involving the**conic**sections were also given by Adrianus Romanus, Vieta, Newton and others.The generality of treatment is indeed remarkable; he gives as the fundamental property of all the conics the equivalent of the Cartesian equation referred to oblique axes (consisting of a diameter and the tangent at its extremity) obtained by cutting an oblique circular cone in any manner, and the axes appear only as a particular case after he has shown that the property of the

**conic**can be expressed in the same form with reference to any new diameter and the tangent at its extremity.AdvertisementIn one solution of the former problem is the first recorded use of the property of a

**conic**(a hyperbola) with reference to the focus and directrix.The first three books of this treatise were translated into English, and several times printed as The Elements of the

**Conic**Sections.There are wonderful stories on record of his precocity in mathematical learning, which is sufficiently established by the well-attested fact that he had completed before he was sixteen years of age a work on the

**conic**sections, in which he had laid down a series of propositions, discovered by himself, of such importance that they may be said to form the foundations of the modern treatment of that subject.In this way he established the famous theorem that the intersections of the three pairs of opposite sides of a hexagon inscribed in a

**conic**are collinear.For the subjects under this heading see the articles

**CONIC**SECTIONS; CIRCLE; CURVE; GEOMETRICAL CONTINUITY; GEOMETRY, Axioms of; GEOMETRY, Euclidean; GEOMETRY, Projective; GEOMETRY, Analytical; GEOMETRY, Line; KNOTS, MATHEMATICAL THEORY OF; MENSURATION; MODELS; PROJECTION; Surface; Trigonometry.AdvertisementThe meteors, whatever their dimensions, must have motions around the sun in obedience to the law of gravitation in the same manner as planets and comets - that is, in

**conic**sections of which the sun is always at one focus.Apollonius' genius takes its highest flight in Book v., where he treats of normals as minimum and maximum straight lines drawn from given points to the curve (independently of tangent properties), discusses how many normals can be drawn from particular points, finds their feet by construction, and gives propositions determining the centre of curvature at any point and leading at once to the Cartesian equation of the evolute of any

**conic**.The general relations between the parabola, ellipse and hyperbola are treated in the articles Geometry, Analytical, and

**Conic**Sections; and various projective properties are demonstrated in the article Geometry, Projective.Try = o to be a parabola is lbc+mca+nab = o, and the

**conic**for which the triangle of reference is self-conjugate la 2 +143 2 +n7 2 =o is a 2 inn--+b 2 nl+c 2 lm=o.The various forms in areal co-ordinates may be derived from the above by substituting Xa for 1, µb for m and vc for n, or directly by expressing the condition for tangency of the line x+y+z = o to the

**conic**expressed in areal coordinates.AdvertisementSee the bibliography to the articles

**Conic**Sections; Geometry, Analytical; and Geometry, Projective.James Gregory, in his Optica Promota (1663), discusses the forms of images and objects produced by lenses and mirrors, and shows that when the surfaces of the lenses or mirrors are portions of spheres the images are curves concave towards the objective, but if the curves of the surfaces are

**conic**sections the spherical aberration is corrected.Thus Whewell mistook Kepler's inference that Mars moves in an ellipse for an induction, though it required the combination of Tycho's and Kepler's observations, as a minor, with the laws of

**conic**sections discovered by the Greeks, as a major, premise.If the roof be not horizontal, we may obtain in this way any form of

**conic**section.Now in a

**conic**whose focus is at 0 we have where 1 is half the latus-rectum, a is half the major axis, and the upper or lower sign is to be taken according as the**conic**is an ellipse or hyperbola.This is recognized as the polar equation of a

**conic**referred to the focus, the half latus-rectum being hf/u.Thus for a circular orbit with the centre of force at an excentric point, the hodograph is a

**conic**with the pole as focus.Hence the meridian section of the film may be traced by the focus of such a

**conic**, if the**conic**is made to roll on the axis.When the

**conic**is an ellipse the meridian line is in the form of a series of waves, and the film itself has a series of alternate swellings and contractions as represented in figs.In all these cases the internal pressure exceeds the external by 2T/a where a is the semi-transverse axis of the

**conic**. The resultant of the internal pressure and the surface-tension is equivalent to a tension along the axis, and the numerical value of this tension is equal to the force due to the action of this pressure on a circle whose diameter is equal to the conjugate axis of the ellipse.In the first volume Of the Entwickelungen he applied the method of abridged notation to the straight line, circle and

**conic**sections, and he subsequently used it with great effect in many of his researches, notably in his theory of cubic curves.As Apollo Agyieus he was shown by a simple

**conic**pillar; the Apollo of Amyclae was a pillar of bronze surmounted by a helmeted head, with extended arms carrying lance and bow.A curve of the second order is a

**conic**, and is also called a quadric curve; and conversely every**conic**is a curve of the second order or quadric curve.Stating the theorem in regard to a

**conic**, we have a real point P (called the pole) and a real line XY (called the polar), the line joining the two (real or imaginary) points of contact of the (real or imaginary) tangents drawn from the point to the**conic**; and the theorem is that when the point describes a line the line passes through a point, this line and point being polar and pole to each other.The points in question have since been called (it is believed first by Dr George Salmon) the circular points at infinity, or they may be called the circular points; these are also frequently spoken of as the points I, J; and we have thus the circle characterized as a

**conic**which passes through the two circular points at infinity; the number of conditions thus imposed upon the**conic**is = 2, and there remain three arbitrary constants, which is the right number for the circle.Secondly, as to the inflections, the process is a similar one; it can be shown that the inflections are the intersections of the curve by a derivative curve called (after Ludwig Otto Hesse who first considered it) the Hessian, defined geometrically as the locus of a point such that its

**conic**polar (§ 8 below) in regard to the curve breaks up into a pair of lines, and which has an equation H = o, where H is the determinant formed with the second differential coefficients of u in regard to the variables (x, y, z); H= o is thus a curve of the order 3 (m - 2), and the number of inflections is =3m(m-2).By means of Pliicker's equations we may form a table - The table is arranged according to the value of in; and we have m=o, n= r, the point; m =1, n =o, the line; m=2, n=2, the

**conic**; of m = 3, the cubic, there are three cases, the class being 6, 4 or 3, according as the curve is without singularities, or as it has 1 node or r cusp; and so of m =4, the quartic, there are ten cases, where observe that in two of them the class is = 6, - the reduction of class arising from two cusps or else from three nodes.The epithets hyperbolic and parabolic are of course derived from the

**conic**hyperbola and parabola respectively.The two legs of a parabolic branch may converge to ultimate parallelism, as in the

**conic**parabola, or diverge to ultimate parallelism, as in the semi-cubical parabola y 2 = x 3, and the branch is said to be convergent, or divergent, accordingly; or they may tend to parallelism in opposite senses, as in the cubical parabola y = x 3 .A circuit is met by any right line always in an even number, or always in an odd number, of points, and it is said to be an even circuit or an odd circuit accordingly; the right line is an odd circuit, the

**conic**an even circuit.We may consider in relation to a curve, not only the line infinity, but also the circular points at infinity; assuming the curve to be real, these present themselves always conjointly; thus a circle is a

**conic**passing through the two circular points, and is thereby distinguished from other conics.A line became continuous, returning into itself by way of infinity; two parallel lines intersect in a point at infinity; all circles pass through two fixed points at infinity (the circular points); two spheres intersect in a fixed circle at infinity; an asymptote became a tangent at infinity; the foci of a

**conic**became the intersections of the tangents from the circular points at infinity; the centre of a**conic**the pole of the line at infinity, &c. In analytical geometry the line at infinity plays an important part in trilinear co-ordinates.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.One definition, which is of especial value in the geometrical treatment of the

**conic**sections (ellipse, parabola and hyperbola) in piano, is that a**conic**is the locus of a point whose distances from a fixed point (termed the focus) and a fixed line (the directrix) are in constant ratio.This ratio, known as the eccentricity, determines the nature of the curve; if it be greater than unity, the

**conic**is a hyperbola; if equal to unity, a parabola; and if less than unity, an ellipse.In the case of the circle, the centre is the focus, and the line at infinity the directrix; we therefore see that a circle is a

**conic**of zero eccentricity.In projective geometry it is convenient to define a

**conic**section as the projection of a circle.A

**conic**may also be regarded as the polar reciprocal of a circle for a point; if the point be without the circle the**conic**is an ellipse, if on the circle a parabola, and if within the circle a hyperbola.In analytical geometry the

**conic**is represented by an algebraic equation of the second degree, and the species of**conic**is solely determined by means of certain relations between the coefficients.The definitions given above reflect the intimate association of these curves, but it frequently happens that a particular

**conic**is defined by some special property (as the ellipse, which is the locus of a point such that the sum of its distances from two fixed points is constant); such definitions and other special properties are treated in the articles Ellipse, Hyperbola and Parabola.The invention of the

**conic**sections is to be assigned to the school of geometers founded by Plato at Athens about the 4th century B.C. Under the guidance and inspiration of this philosopher much attention was given to the geometry of solids, and it is probable that while investigating the cone, Menaechrnus, an associate of Plato, pupil of Eudoxus, and brother of Dinostratus (the inventor of the quadratrix), discovered and investigated the various curves made by truncating a cone.But the greatest Greek writer on the

**conic**sections was Apollonius of Perga, and it is to his**Conic**Sections that we are indebted for a review of the early history of this subject.The first book, which is almost entirely concerned with the construction of the three

**conic**sections, contains one of the most brilliant of all the discoveries of Apollonius.Prior to his time, a right cone of a definite vertical angle was required for the generation of any particular

**conic**; Apollonius showed that the sections could all be produced from one and the same cone, which may be either right or oblique, by simply varying the inclination of the cutting plane.The fifth book contains properties of normals and their envelopes, thus embracing the germs of the theory of evolutes, and also maxima and minima problems, such as to draw the longest and shortest lines from a given point to a

**conic**; the sixth book is concerned with the similarity of conics; the seventh with complementary chords and conjugate diameters; the eighth book, according to the restoration of Edmund Halley, continues the subject of the preceding book.His proofs are generally long and clumsy; this is accounted for in some measure by the absence of symbols and technical terms. Apollonius was ignorant of the directrix of a

**conic**, and although he incidentally discovered the focus of an ellipse and hyperbola, he does not mention the focus of a parabola.This work, the earliest published in Christian Europe, treats the

**conic**sections in relation to the original cone, the procedure differing from that of the Greek geometers.Since all conics derived from a circular cone appear circular when viewed from the apex, they conceived the treatment of the

**conic**sections as projections of a circle.Desargues has a special claim to fame on account of his beautiful theorem on the involution of a quadrangle inscribed in a

**conic**. Pascal discovered a striking property of a hexagon inscribed in a**conic**(the hexagrammum mysticum); from this theorem Pascal is said to have deduced over 400 corollaries, including most of the results obtained by earlier geometers.The subject of analytical geometry which he virtually created enabled him to view the

**conic**sections as algebraic equations of the second degree, the form of the section depending solely on the coefficients.John Wallis, in addition to translating the Conics of Apollonius, published in 1655 an original work entitled De sectionibus conicis nova methodo expositis, in which he treated the curves by the Cartesian method, and derived their properties from the definition in piano, completely ignoring the connexion between the

**conic**sections and a cone.Philippe de la Hire, a pupil of Desargues, wrote several works on the

**conic**sections, of which the most important is his Sectiones Conicae (1685).In Newton's method, two angles of constant magnitude are caused to revolve about their vertices which are fixed in position, in such a manner that the intersection of two limbs moves along a fixed straight line; then the two remaining limbs envelop a

**conic**. Maclaurin's method, published in his Geometria organica (1719), is based on the proposition that the locus of the vertex of a triangle, the sides of which pass through three fixed points, and the base angles move along two fixed lines, is a**conic**section.In modern times the study of the

**conic**sections has proceeded along the lines which we have indicated; for further details reference should be made to the article Geometry.A copious list of early works on

**conic**sections is given in Fred.Eagles, Constructive Geometry of Plane Curves (1886); geometric investigations primarily based on the relation of the

**conic**sections to a cone are given in Hugo Hamilton's De Sectionibus Conicis (1758); this method of treatment has been largely replaced by considering the curves from their definition in piano, and then passing to their derivation from the cone and cylinder.Mar Burn Beyond Burn of Mar the path climbs fairly quickly along the northern flank of

**Conic**Hill.Silver Pyramid Black Cat's

**conic**fountain creates a surprisingly high pillar of sparks.Analytically, it is defined by an equation of the second degree, of which the highest terms have real roots (see

**Conic**Section).If the law of attraction is that of gravitation, the orbit is a

**conic**section - ellipse, parabola or hyperbola - having the centre of attraction in one of its foci; and the motion takes place in accordance with Kepler's laws (see Astronomy).The Arabians more closely resembled the Hindus than the Greeks in the choice of studies; their philosophers blended speculative dissertations with the more progressive study of medicine; their mathematicians neglected the subtleties of the

**conic**sections and Diophantine analysis, and applied themselves more particularly to perfect the system of numerals (see Numeral), arithmetic and astronomy.The equation to a parabola in triangular co-ordinates is generally derived by expressing the condition that the line at infinity is a tangent in the equation to the general

**conic**. For example, in trilinear co-ordinates, the equation to the general**conic**circumscribing the triangle of reference is 113y+mya+naf3=o; for this to be a parabola the line as + b/ + cy = o must be a tangent.On this is based the great structure of celestial mechanics and the theory of universal gravitation; and in the elucidation of problems more directly concerned with astronomy, Kepler, Sir Isaac Newton and others discovered many properties of the

**conic**sections (see Mechanics).An important generalization of the

**conic**sections was developed about the beginning of the 17th century by Girard Desargues and Blaise Pascal.The comprehensive scheme of study included mathematics also, in which he advanced as far as the

**conic**sections in the treatise of L'Hopital.After his Tables of the Products and Powers of Numbers, 1781, and his Mathematical Tables, 1785, he issued, for the use of the Royal Military Academy, in 1787 Elements of

**Conic**Sections, and in 1798 his Course of Mathematics.