# Conics Sentence Examples

- We may now summarize the contents of the
**Conics**of Apollonius. - In it Maclaurin developed several theorems due to Newton, and introduced the method of generating
**conics**which bears his name, and showed that many curves of the third and fourth degrees can be described by the intersection of two movable angles. - Astronomy was also enriched by his investigations, and he was led to several remarkable theorems on
**conics**which bear his name. - His treatise on
**Conics**gained him the title of The Great Geometer, and is that by which his fame has been transmitted to modern times. - After the
**Conics**in eight Books had been written in a first edition, Apollonius brought out a second edition, considerably revised as regards Books i.-ii., at the instance of one Eudemus of Pergamum; the first three books were sent to Eudemus at intervals, as revised, and the later books were dedicated (after Eudemus' death) to King Attalus I. - The degree of originality of the
**Conics**can best be judged from Apollonius' own prefaces. - That he made the fullest use of his predecessors' works, such as Euclid's four Books on
**Conics**, is clear from his allusions to Euclid, Conon and Nicoteles. - 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. - Each of these was divided into two books, and, with the Data, the Porisms and Surface-Loci of Euclid and the
**Conics**of Apollonius were, according to Pappus, included in the body of the ancient analysis. - (this is the monumental edition of Edmund Halley); (3) the edition of the first four books of the
**Conics**given in 1675 by Barrow; (4) Apollonii Pergaei de Sectione Rationis libri duo: Accedunt ejusdem de Sectione Spatii libri duo Restituti: Praemittitur, e g c., Opera et Studio Edmundi Halley (Oxoniae, 1706), 4to; (5) a German translation of the**Conics**by H. - This proposition, which he called the mystic hexagram, he made the keystone of his theory; from it alone he deduced more than 400 corollaries, embracing, according to his own account, the
**conics**of Apollonius, and other results innumerable. - To these should be added his version from the Arabic (which language he acquired for the purpose) of the treatise of Apollonius De sectione rationis, with a restoration of his two lost books De sectione spatii, both published at Oxford in 1706; also his fine edition of the
**Conics**of Apollonius, with the treatise by Serenus De sectione cylindri et coni (Oxford, 1710, folio). - Hypatia, according to Suidas, was the author of commentaries on the Arithmetica of Diophantus of Alexandria, on the
**Conics**of Apollonius of Perga and on the astronomical canon (of Ptolemy). - Conjointly with Giovanni Borelli he wrote a Latin translation of the 5th, 6th and 7th books of the
**Conics**of Apollonius of Perga (1661). - Thus the curve of the first order or right line consists of one branch; but in curves of the second order, or
**conics**, the ellipse and the parabola consist each of one branch, the hyperbola of two branches. - The most simple example is in the two systems of equations x': y': z' = yz: zx: xy and x: z'x': x'y'; where yz =0, zx =0, xy = o are
**conics**(pairs of lines) having three common intersections, and where obviously either system of equations leads to the other system. - Coming next to Chasles, the principle of correspondence is established and used by him in a series of memoirs relating to the
**conics**which satisfy given conditions, and to other geometrical questions, contained in the Comptes rendus, t. - The principle in its original form as applying to a right line was used throughout by Chasles in the investigations on the number of the
**conics**which satisfy given conditions, and on the number of solutions of very many other geometrical problems. - Lviii., 1864, et seq.) refer to the
**conics**which satisfy given conditions. - But Chasles in the first of his papers (February 1864), considering the
**conics**which satisfy four conditions, establishes the notion of the two characteristics (µ, v) of such a system of**conics**, viz. - ,u is the number of the
**conics**which pass through a given arbitrary point, and v is the number of the**conics**which touch a given arbitrary line. - And he gives the theorem, a system of
**conics**satisfying four conditions, and having the characteristics (µ, v) contains 2v - µ line-pairs (that is,**conics**, each of them a pair of lines), and point-pairs (that is,**conics**, each of them a pair of points, - coniques infiniment aplaties), which is a fundamental one in the theory. - The characteristics of the system can be determined when it is known how many there are of these two kinds of degenerate
**conics**in the system, and how often each is to be counted. - It was thus that Zeuthen (in the paper Nyt Bydrag, " Contribution to the Theory of Systems of
**Conics**which satisfy four Conditions " (Copenhagen, 1865), translated with an addition in the Nouvelles Annales) solved the question of finding the characteristics of the systems of**conics**which satisfy four conditions of contact with a given curve or curves; and this led to the solution of the further problem of finding the number of the**conics**which satisfy five conditions of contact with a given curve or curves (Cayley, Comptes Rendus, t. - It was thus that Zeuthen (in the paper Nyt Bydrag, " Contribution to the Theory of Systems of
**Conics**which satisfy four Conditions " (Copenhagen, 1865), translated with an addition in the Nouvelles Annales) solved the question of finding the characteristics of the systems of**conics**which satisfy four conditions of contact with a given curve or curves; and this led to the solution of the further problem of finding the number of the**conics**which satisfy five conditions of contact with a given curve or curves (Cayley, Comptes Rendus, t. - It may be remarked that although, as a process of investigation, it is very convenient to seek for the characteristics of a system of
**conics**satisfying 4 conditions, yet what is really determined is in every case the number of the**conics**which satisfy 5 conditions; the characteristics of the system (4 p) of the**conics**which pass through 4p points are (5 p), (4 p, il), the number of the**conics**which pass through 5 points, and which pass through 4 points and touch 1 line: and so in other cases. - Metrical relations between the axes, eccentricity, distance between the foci, and between these quantities and the co-ordinates of points on the curve (referred to the axes and the centre), and focal distances are readily obtained by the methods of geometrical
**conics**or analytically. - The circle, and two lines (and also two points, the reciprocal of two lines) under the general title conic. The definition of
**conics**as sections of a cone was employed by the Greek geometers as the fundamental principle of their researches in this subject; but the subsequent development of geometrical methods has brought to light many other means for defining these curves. - Confocal
**conics**are**conics**having the same foci. - If one of the foci be at infinity, the
**conics**are confocal parabolas, which may also be regarded as parabolas having a common focus and axis. - In this article we shall consider the historical development of the geometry of
**conics**, and refer the reader to the article Geometry: Analytical and Projective, for the special methods of investigation. - On the authority of the two great commentators Pappus and Proclus, Euclid wrote four books on
**conics**, but the originals are now lost, and all we have is chiefly to be found in the works of Apollonius of Perga. - He probably wrote a book on
**conics**, but it is now lost. - He discriminated the three species of
**conics**as follows: - At one of the two vertices erect a perpendicular (talus rectum) of a certain length (which is determined below), and join the extremity of this line to the other vertex. - This property is true for all
**conics**, and it served as the basis of most of the constructions and propositions given by Apollonius. - The
**conics**are distinguished by the ratio between the latus rectum (which was originally called the latus erectum, and now often referred to as the parameter) and the segment of the ordinate intercepted between the diameter and the line joining the second vertex with the extremity of the latus rectum. - The first book deals with the generation of the three
**conics**; the second with the asymptotes, axes and diameters; the third with various metrical relations between transversals, chords, tangents, asymptotes, &c.; the fourth with the theory of the pole and polar, including the harmonic division of a straight line, and with systems of two**conics**, which he shows to intersect in not more than four points; he also investigates**conics**having single and double contact. - 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. - The
**Conics**of Apollonius was translated into Arabic by Tobit ben Korra in the 9th century, and this edition was followed by Halley in 1710. - Although the Arabs were in full possession of the store of knowledge of the geometry of
**conics**which the Greeks had accumulated, they did little to increase it; the only advance made consisted in the application of describing intersecting**conics**so as to solve algebraic equations. - In 1522 there was published an original work on
**conics**by Johann Werner of Nuremburg. - 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. - From this conception all the properties of
**conics**can be deduced. - 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. - A mathematical investigation of the
**conics**by this method is given in the article Geometry: Analytical. - His treatment is synthetic, and he follows his tutor and Pascal in deducing the properties of
**conics**by projection from a circle. - A method of generating
**conics**essentially the same as our modern method of homographic pencils was discussed by Jan de Witt in his Elementa linearum curvarum (1650); but he treated the curves by the Cartesian method, and not synthetically. - Michel Chasles in his Apercu historique sur l'origine et le de'veloppement des methodes en geometrie (1837, a third edition was published in 1889), gives a valuable account of both the ancient and modern geometry of
**conics**; a German translation with the title Geschichte der Geometrie was published in 1839 by L. - Besant, Geometrical Conic Sections; C. Smith, Geometrical
**Conics**; W. - Reference may also be made to C. Taylor, An Introduction to Ancient and Modern Geometry of
**Conics**(1881). - He next gives by aid of these projective rows and pencils a new generation of
**conics**and ruled quadric surfaces, "which leads quicker and more directly than former methods into the inner nature of**conics**and reveals to us the organic connexion of their innumerable properties and mysteries."