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.
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 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).Advertisement
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.
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.
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.Advertisement
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.
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.
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.
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.Advertisement
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.
That he made considerable progress in the study of these curves is evidenced by Eutocius, who flourished about the 6th century A.D., and who assigns to Menaechmus two solutions of the problem of duplicating the cube by means of intersecting conics.
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.Advertisement
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.
Johann Kepler (1571-1630) made many important discoveries in the geometry of conics.
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.
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.
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."
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.