Conic-sections Sentence Examples

The comprehensive scheme of study included mathematics also, in which he advanced as far as the conic sections in the treatise of L'Hopital.

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.

Cubic equations were solved geometrically by determining the intersections of conic sections.

He improved the methods for solving equations, and devised geometrical constructions with the aid of the conic sections.

The 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.

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.

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.

The first three books of this treatise were translated into English, and several times printed as The Elements of the Conic Sections.

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.

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.

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.

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.

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.

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.

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.

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.