This website uses cookies to ensure you get the best experience. Learn more

geometry

I think we shared a geometry class the last year of school.

195108For example, geometry is such a department.

11970Now geometry deals with points, lines, planes and cubic contents.

5822His life was devoted to the study of higher geometry and reforming the more advanced mathematical teaching of Italy.

4628The geometry of the rectangular hyperbola is simplified by the fact that its principal axes are equal.

4123One paper has Helen demonstrating problems in geometry by means of her playing blocks.

3722The "axioms" of geometry are the fixed conditions which occur in the hypotheses of the geometrical propositions.

3316He was elected fellow of Balliol in 1850 and Savilian professor of geometry in 1861, and in 1874 was appointed keeper of the university museum.

2814The Canonis Descriptio on its publication in 1614, at once attracted the attention of Edward Wright, whose name is known in connexion with improvements in navigation, and Henry Briggs, then professor of geometry at Gresham College, London.

2712The Greeks created the sciences of geometry and of number as applied to the measurement of continuous quantities.

2120Thus Descartes gave to modern geometry that abstract and general character in which consists its superiority to the geometry of the ancients.

209However, the braille worked well enough in the languages; but when it came to Geometry and Algebra, it was different.

1916In fact, the whole theory of measurement in geometry arises at a comparatively late stage as the result of a variety of complicated considerations.

188Fortunately, however, Robert Napier had transcribed his father's manuscript De Arte Logistica, and the copy escaped the fate of the originals in the manner explained in the following note, written in the volume containing them by Francis, seventh Lord Napier: "John Napier of Merchiston, inventor of the logarithms, left his manuscripts to his son Robert, who appears to have caused the following pages to have been written out fair from his father's notes, for Mr Briggs, professor of geometry at Oxford.

189The book will contain four essays, all in French, with the general title of Project of a Universal science, capable of raising our nature to its highest perfection; also Dioptrics, Meteors and Geometry, wherein the most curious matters which the author could select as a proof of the universal science which he proposes are explained in such a way that even the unlearned may understand them.'

169In mathematics, he was the first to draw up a methodical treatment of mechanics with the aid of geometry; he first distinguished harmonic progression from arithmetical and geometrical progressions.

1611During this period logarithms were invented, trigonometry and algebra developed, analytical geometry invented, dynamics put upon a sound basis, and the period closed with the magnificent invention of (or at least the perfecting of) the differential calculus by Newton and Leibnitz and the discovery of gravitation.

1611In 1664 Sir John Cutler instituted for his benefit a mechanical lectureship of £50 a year, and in the following year he was nominated professor of geometry in Gresham College, where he subsequently resided.

149Algebra and geometry were the only studies that continued to defy my efforts to comprehend them.

1412The Geometry of Descartes, unlike the other parts of his essays, is not easy reading.

1211The ancient geometry, as we know it, is a wonderful monument of ingenuity - a series of tours de force, in which each problem to all appearance stands alone, and, if solved, is solved by methods and principles peculiar to itself.

118Consequently, I did not do so well as I should have done, if Teacher had been allowed to read the Algebra and Geometry to me.

119He took the exercise book containing lessons in geometry written by himself and drew up a chair with his foot.

119In 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.

1111She put down the geometry book and eagerly broke the seal of her letter.

1010Thus what have been called seminvariants are not all of them invariants for the general substitution, but are invariants for the particular substitution xl = X11 + J-s12, X 2 = 112 Again, in plane geometry, the most general equations of substitution which change from old axes inclined at w to new axes inclined at w' =13 - a, and inclined at angles a, l3 to the old axis of x, without change of origin, are x-sin(wa)X+sin(w -/3)Y sin w sin ' _sin ax y sin w a transformation of modulus sin w' sin w' The theory of invariants originated in the discussion, by George Boole, of this system so important in geometry.

95His manual on Graphical Statics and his Elements of Projective Geometry (translated by C. Leudesdorf), have been published in English by the Clarendon Press.

96Also, as the Cartesian geometry shows, all the relations between points are expressible in terms of geometric quantities.

97Geometry again is regarded by thoroughgoing empiricists as hypothetical.

98Also in geometry, what is a point ?

98But as yet he had only glimpses of a logical method which should invigorate the syllogism by the co-operation of ancient geometry and modern algebra.

85But the pupil soon found his teacher to be a charlatan, and taught himself, aided by commentaries, to master logic, geometry and astronomy.

85The braille worked well enough in the languages, but when it came to geometry and algebra, difficulties arose.

86It is true that I was familiar with all literary braille in common use in this country--English, American, and New York Point; but the various signs and symbols in geometry and algebra in the three systems are very different, and I had used only the English braille in my algebra.

86Fermat, Roberval and Desargues took exception in their various ways to the methods employed in the geometry, and to the demonstrations of the laws of refraction given in the Dioptrics and Meteors.

87The three subjects to which Smith's writings relate are theory of numbers, elliptic functions and modern geometry; but in all that he wrote an "arithmetical" made of thought is apparent, his methods and processes being arithmetical as distinguished from algebraic. He had the most intense admiration of Gauss.

87In 1860 he was appointed to the professorship of higher geometry at the university of Bologna, and in 1866 to that of higher geometry and graphical statics at the higher technical college of Milan.

810While at Oxford Wren distinguished himself in geometry and applied mathematics, and Newton, in his Principia, p. 19 (ed.

76Between Roberval and Descartes there existed a feeling of ill - will, owing to the jealousy aroused in the mind of the former by the criticism which Descartes offered to some of the methods employed by him and by Pierre de Fermat; and this led him to criticize and oppose the analytical methods which Descartes introduced into geometry about this time.

76Thus the whole method of measurement in geometry as described in the elementary textbooks and the older treatises is obscure to the last degree.

76Accordingly, at first sight it seems reasonable to define geometry in some such way as "the science of dimensional quantity."

77"The hours are the same, and the lathe, and also the mathematics and my geometry lessons," said Princess Mary gleefully, as if her lessons in geometry were among the greatest delights of her life.

77His earliest publications, beginning with A Syllabus of Plane Algebraical Geometry (1860) and The Formulae of Plane Trigonometry (1861), were exclusively mathematical; but late in the year 1865 he published, under the pseudonym of "Lewis Carroll," Alice's Adventures in Wonderland, a work that was the outcome of his keen sympathy with the imagination of children and their sense of fun.

78The manuscripts of the geometry of Boetius differ widely from each other.

65This work entitles Poncelet to rank as one of the greatest of those who took part in the development of the modern geometry of which G.

65In addition to the various works of Brewster already noticed, the following may be mentioned: - Notes and Introduction to Carlyle's translation of Legendre's Elements of Geometry (1824); Treatise on Optics (1831); Letters on Natural Magic, addressed to Sir Walter Scott (1831); The Martyrs of Science, or the Lives of Galileo, Tycho Brake, and Kepler (1841); More Worlds than One (1854).

65This fruitful thought he illustrates by showing how geometry is applied to the action of natural bodies, and demonstrating by geometrical figures certain laws of physical forces.

55In 1709 he entered the university of Glasgow, where he exhibited a decided genius for mathematics, more especially for geometry; it is said that before the end of his sixteenth year he had discovered many of the theorems afterwards published in his Geometria organica.

55From February to July, 1898, Mr. Keith came out to Wrentham twice a week, and taught me algebra, geometry, Greek and Latin.

55Allman, Greek Geometry from Thales to Euclid (1889); Florian Cajori, History of Mathematics (New York, 1894); M.

56But he seems to have been well cared for, and he was at the age of fourteen sufficiently advanced "in algebra, geometry, astronomy, and even the higher mathematics," to calculate a solar eclipse within four seconds of accuracy.

56His writings include: Mathematical Investigations in the Theory of Value and Prices (1892); Elements of Geometry (with A.

57de Traytorrens, went through the elements of algebra and geometry, and the three fi r st books of the Marquis de l'Hopital's Conic Sections.

47Between them the general theory of the complex variable, and of the various "infinite" processes of mathematical analysis, was established, while other mathematicians, such as Poncelet, Steiner, Lobatschewsky and von Staudt, were founding modern geometry, and Gauss inaugurated the differential geometry of surfaces.

412The special nature of the "axioms" which constitute geometry is considered in the article Geometry (Axioms).

310It was during his imprisonment here that, "prive de toute espece de livres et de secours, surtout distrait par les malheurs de ma patrie et les miens propres," as he himself puts it, he began his researches on projective geometry which led to his great treatise on that subject.

315The first day I had Elementary Greek and Advanced Latin, and the second day Geometry, Algebra and Advanced Greek.

211Mobius must be regarded as one of the leaders in the introduction of the powerful methods of modern projective geometry.

00Having thus perfected the instrument, his next step was to apply it in such a way as to bring uniformity of method into the isolated and independent operations of geometry.

00Such is the basis of the algebraical or modern analytical geometry.

00The coordinates thus found will in the case of a body moving around the sun be heliocentric. The reduction to the earth's centre is a problem of pure geometry.

00His largest work,Trattato generale di numeri e misure, is a comprehensive mathematical treatise, including arithmetic, geometry, mensuration, and algebra as far as quadratic equations (Venice, 1556, 1560).

00Geometry.

00Under the general heading "Geometry" occur the subheadings "Foundations," with the topics principles of geometry, non-Euclidean geometries, hyperspace, methods of analytical geometry; "Elementary Geometry," with the topics planimetry, stereometry, trigonometry, descriptive geometry; "Geometry of Conics and Quadrics," with the implied topics; "Algebraic Curves and Surfaces of Degree higher than the Second," with the implied topics; "Transformations and General Methods for Algebraic Configurations," with the topics collineation, duality, transformations, correspondence, groups of points on algebraic curves and surfaces, genus of curves and surfaces, enumerative geometry, connexes, complexes, congruences, higher elements in space, algebraic configurations in hyperspace; "Infinitesimal Geometry: applications of Differential and Integral Calculus to Geometry," with the topics kinematic geometry, curvature, rectification and quadrature, special transcendental curves and surfaces; "Differential Geometry: applications of Differential Equations to Geometry," with the topics curves on surfaces, minimal surfaces, surfaces determined by differential properties, conformal and other representation of surfaces on others, deformation of surfaces, orthogonal and isothermic surfaces.

00For 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.

00These headings are: "Geometry and Kinematics of Particles and Solid Bodies"; "Principles of Rational Mechanics"; "Statics of Particles, Rigid Bodies, &c."; "Kinetics of Particles, Rigid Bodies, &c."; "General Analytical Mechanics"; "Statics and Dynamics of Fluids"; "Hydraulics and Fluid Resistances"; "Elasticity."

00Arithmetical groups, connected with the theory of quadratic forms and other branches of the theory of numbers, which are termed "discontinuous," and infinite groups connected with differential forms and equations, came into existence, and also particular linear and higher transformations connected with analysis and geometry.

00His earliest tastes were literary rather than scientific, and he learned the rudiments of geometry during his first year at the college of Turin, without difficulty, but without distinction.

00Appointed, in 1754, professor of geometry in the royal school of artillery, he formed with some of his pupils - for the most part his seniors - friendships based on community of scientific ardour.

00application to geometry, and the third with its bearings on mechanics.

00On the establishment of the Institute, Lagrange was placed at the head of the section of geometry; he was one of the first members of the Bureau des Longitudes; and his name appeared in 1791 on the list of foreign members of the Royal Society.

00It was his just boast to have transformed mechanics (defined by him as a "geometry of four dimensions") into a branch of analysis, and to have exhibited the so-called mechanical "principles" as simple results of the calculus.

00This reaction has taken the form of a return to the alliance between algebra and geometry (ï¿½5), on which modern analytical geometry is based; the alliance, however, being concerned with the application of graphical methods to particular cases rather than to general expressions.

00The progress of analytical geometry led to a geometrical interpretation both of negative and also of imaginary quantities; and when a " meaning " or, more properly, an interpretation, had thus been found for the symbols in question, a reconsideration of the old algebraic problem became inevitable, and the true solution, now so obvious, was eventually obtained.

00It can be proved by geometry that (aA-H3B) +yC = aA+(aB+- y C) = (a + 1 3+ 7) P, where P is in fact the centroid of masses a, 13, y placed at A, B, C respectively.

00We find that geometry was neglected except in so far as it was of service to astronomy; trigonometry was advanced, and algebra improved far beyond the attainments of Diophantus.

00This author questioned the possibility of solving cubics by pure algebra, and biquadratics by geometry.

00This was Lucas Paciolus (Lucas de Burgo), a Minorite friar, who, having previously written works on algebra, arithmetic and geometry, published, in 1494, his principal work, entitled Summa de Arithmetica, Geometria, Proportioni et Proportionalita.

00So far the development of algebra and geometry had been mutually independent, except for a few isolated applications of geometrical constructions to the solution of algebraical problems. Certain minds had long suspected the advantages which would accrue from the unrestricted application of algebra to geometry, but it was not until the advent of the philosopher Rene Descartes that the co-ordination was effected.

00A long treatise on geometry, attributed to Gerbert, is of somewhat doubtful authenticity.

00He also studied philosophy, astronomy and geometry, and wrote works on those subjects, which, together with his consulship, formed the subject of a panegyric by Claudian.

00At the same time, it delights the pure theorist by the simplicity of the logic with which the fundamental theorems may be established, and by the elegance of its mathematical operations, insomuch that hydrostatics may be considered as the Euclidean pure geometry of mechanical science.

00is given up to mathematics, under which head are included music, geometry, astronomy, astrology, weights and measures, and metaphysics.

00Comte's series or hierarchy is arranged as follows: (i) Mathematics (that is, number, geometry, and mechanics), (2) Astronomy, (3) Physics, (4) Chemistry, (5) Biology, (6) Sociology.

00Then it can be shown that I /p2 = x2/a4+y2/b4+z2/c4 (see Frost's Solid Geometry, p. 172).

00He was educated at Balliol College, Oxford, and in 1630 was chosen professor of geometry in Gresham College, London.

00With the marquis de l'Hopital he spent four months studying higher geometry and the resources of the new calculus.

00Meanwhile the study of mathematics was not neglected, as appears not only from his giving instruction in geometry to his younger brother Daniel, but from his writings on the differential, integral, and exponential calculus, and from his father considering him, at the age of twenty-one, worthy of receiving the torch of science from his own hands.

00This study, however, did not check his hereditary taste for geometry.

00Another of his works, Recensio canonica effectionum geometricarum, bears a stamp not less modern, being what we now call an algebraic geometry - in other words, a collection of precepts how to construct algebraic expressions with the use of rule and compass only.

00Sca, through,, u rpov, measure), in geometry, a line passing through the centre of a circle or conic section and terminated by the curve; the "principal diameters of the ellipse and hyperbola coincide with the "axes" and are at right angles; " conjugate diameters " are such that each bisects chords parallel to the other.

00He graduated in arts, and claims to have graduated in medicine (of this there is no record at Paris), published six lectures on " syrups " (the most popular of his works), lectured on geometry and " astrology " (from a medical point of view) and defended by counsel a suit brought against him (March 1538) by the medical faculty on the ground of his astrological lectures.

00The fragments of a work De Natali Institutione, dealing with astronomy, geometry, music and versification, and usually printed with the De Die Natali of Censorinus, are not by him.

00The magnitudes, on the other hand, which we meet with in geometry, are essentially continuous.

00of geometry, vii.

001875); Examples of Analytical Geometry of Three Dimensions (1858, 3rd ed., 1873); Mechanics (1867), History of the Mathematical Theory of Probability from the Time of Pascal to that of Lagrange (1865); Researches in the Calculus of Variations (1871); History of the Mathematical Theories of Attraction and Figure of the Earth from Newton to Laplace (1873); Elementary Treatise on Laplace's, Lame's and Bessel's Functions (1875); Natural Philosophy for Beginners (1877).

00eKovt, twenty, and g (3pa, a face or base), in geometry, a solid enclosed by twenty faces.

00of distances between points) as belonging to geometry or trigonometry; while the measurement of curved lengths, except in certain special cases, involves the use of the integral calculus.

00This use of formulae for dealing with numbers, which express magnitudes in terms of units, constitutes the broad difference between mensuration and ordinary geometry, which knows nothing of units.

00This applies not only to the geometrical principles but also to the arithmetical principles, and it is therefore of importance, in the earlier stages, to keep geometry, mensuration and arithmetic in close association with one another; mensuration forming, in fact, the link between arithmetic and geometry.

00For fuller discussion reference should be made to Geometry and Trigonometry, as well as to the articles dealing with particular figures, such as Triangle, Circle, &C.

00In elementary geometry we deal with lines and curves, while in mensuration we deal with areas bounded by these lines or curves.

00Similarly, analytical plane geometry deals with the curve described by a point moving in a particular way, while analytical plane mensuration deals with the figure generated by an ordinate moving so that its length varies in a particular manner depending on its position.

00In the same way, in the case of a figure in three dimensions, analytical geometry is concerned with the form of the surface, while analytical mensuration is concerned with the figure as a whole.

00Jackson, Elementary Solid Geometry (1907); P. A.

00Millis, Technical Arithmetic and Geometry (1903).

00bisector, from bi-, two, secare, to cut), in geometry, the same as bisector, i.e.

00Kb(30s, a cube), in geometry, a solid bounded by six equal squares, so placed that the angle between any pair of adjacent faces is a right angle.

00This solid played an all-important part in the geometry and cosmology of the Greeks.

00Although Hippocrates could not determine the proportionals, his statement of the problem in this form was a great advance, for it was perceived that the problem of trisecting an angle was reducible to a similar form which, in the language of algebraic geometry, is to solve geometrically a cubic equation.

00The geometry of the sphere was studied by the Greeks; Euclid, in book xii.

00In analytical geometry, the equation to the sphere takes the forms x 2 +y 2 +z 2 =a 2, and r=a, the first applying to rectangular Cartesian co-ordinates, the second to polar, the origin being in both cases at the centre of the sphere.

00The lines in the diagram represent the directions of a series of forces which must all be in equilibrium; these lines may, for an object to be explained in the next paragraph, be conveniently named by the letters in the spaces which they separate instead of by the method usually employed in geometry.

00During forty years the resources of analysis, even in the hands of d'Alembert, Lagrange and Laplace, had not carried the theory of the attraction of ellipsoids beyond the point which the geometry of Maclaurin had reached.

00The Elements of Geometry.

00- Legendre's name is most widely known on account of his Elements de geometrie, the most successful of the numerous attempts that have been made to supersede Euclid as a text-book on geometry.

00Notwithstanding this act of opposition, he was in June 1649 appointed Savilian professor of geometry at Oxford.

00The Mathesis universalis, a more elementary work, contains copious dissertations on fundamental points of algebra, arithmetic and geometry, and critical remarks.

00The De algebra tractatus contains (chapters lxvi.-lxix.) the idea of the interpretation of imaginary quantities in geometry.

00As a mathematician he occupied himself with many branches of his favourite science, more especially with higher algebra, including the theory of determinants, with the general calculus of symbols, and with the application of analysis to geometry and mechanics.

00Henry Briggs, then professor of geometry at Gresham College, London, and afterwards Savilian professor of geometry at Oxford, welcomed the Descriptio with enthusiasm.

00In 1717 Abraham Sharp published in his Geometry Improv'd the Briggian logarithms of numbers from 1 to 100, and of primes from 100 to 1100, to 61 places; these were copied into the later editions of Sherwin and other works.

00The form of a circle is familiar to all; and we proceed to define certain lines, points, &c., which constantly occur in studying its geometry.

00Reference should be made to the article Geometry: Euclidean, for a detailed summary of the Euclidean treatment, and the elementary properties of the circle.

00Analytical Geometry of the Circle.

00In the article Geometry: Analytical, it is shown that the general equation to a circle in rectangular Cartesian co-ordinates is x 2 - { - y 2 + 2gx-}-2fy+c= o, i.e.

00The introduction of these lines and points constitutes a striking achievement in geometry, and from their association with circles they have been named the " circular lines " and " circular points."

00In various systems of triangular co-ordinates the equations to circles specially related to the triangle of reference assume comparatively simple forms; consequently they provide elegant algebraical demonstrations of properties concerning a triangle and the circles intimately associated with its geometry.

00Very early in the history of geometry it was known that the circumference and area of a circle of radius r could be expressed in the forms 27rr and 7rr2.

00The problem of finding a square equal in area to a given circle, like all problems, may be increased in difficulty by the imposition of restrictions; consequently under the designation there may be embraced quite a variety of geometrical problems. It has to be noted, however, that, when the " squaring " of the circle is especially spoken of, it is almost always tacitly assumed that the restrictions are those of the Euclidean geometry.

001 V 2+2A/ From this point onwards, therefore, no knowledge whatever of geometry was necessary in any one who aspired to determine the ratio to any required degree of accuracy; the problem being reduced to an arithmetical computation.

00For in such a construction every point of the figure is obtained by the intersection of two straight lines, a straight line and a circle, or two circles; and as this implies that, when a unit of length is introduced, numbers employed, and the problem transformed into one of algebraic geometry, the equations to be solved can only be of the first or second degree, it follows that the equation to which we must be finally led is a rational equation of even degree.

00number in arithmetic, magnitude in geometry, stars in astronomy, a man's good in ethics; concentrates itself on the causes and appropriate principles of its subject, especially the definition of the subject and its species by their essences or formal causes; and after an inductive intelligence of those principles proceeds by a deductive demonstration from definitions to consequences: philosophy is simply a desire of this definite knowledge of causes and effects.

00Recorde published several works upon mathematical subjects, chiefly in the form of dialogue between master and scholar, viz.: - The Grounde of Artes, teachinge the Worke and Practise of Arithmeticke, both in whole numbers and fractions (1540); The Pathway to Knowledge, containing the First Principles of Geometry.

00FRUSTUM (Latin for a "piece broken off"), a term in geometry for the part of a solid figure, such as a cone or pyramid, cut off by a plane parallel to the base, or lying between two parallel planes; and hence in architecture a name given to the drum of a column.

00Interspersed are some questions of pure geometry.

00He was taught the use of the astrolabe (which Prince Yakov Dolgoruki, with intent to please, had brought him from Paris) by a Dutchman, Franz Timmerman, who also instructed him in the rudiments of geometry and fortifications.

00One while he devoted himself to the sciences, " perfecting himself in music, arithmetic, geometry and ' Life, P. 93.

00The king frequently sent for him into his closet, and discoursed with him on astronomy, geometry and points of divinity.

00In More's house you would see that Plato's Academy was revived again, only, whereas in the Academy the discussions turned upon geometry and the power of numbers, the house at Chelsea is a veritable school of Christian religion.

00Having studied law at Leipzig, Helmstadt and Jena, and mathematics, especially geometry and mechanics, at Leiden, he visited France and England, and in 1636 became engineer-in-chief at Erfurt.

00evyKOs, a bend; both connected with the Aryan root ank-, to bend: see Angling), in geometry, the inclination of one line or plane to another.

00A discussion of these concepts and the various definitions of angles in Euclidean geometry is to be found in W.

00It is compiled out of an Adversaria, or commonplace book, in which he had jotted down everything of unusual interest that he heard in conversation or read in books, and it comprises notes on grammar, geometry, philosophy, history and almost every other branch of knowledge.

00Towards the close of 1794, when the Ecole Polytechnique was established, he was appointed along with Monge over the department of descriptive geometry.

00His labours were chiefly in the field of descriptive geometry, with its application to the arts and mechanical engineering.

00It was left to him to develop the geometry of Monge, and to him also is due in great measure the rapid advancement which France made soon after the establishment of the Ecole Polytechnique in the construction of machinery .

00This theory, however superficial from the standpoint of observation, indicates considerable knowledge of geometry and gave a great impulse to the study of the science.

00THALES OF MILETUS (6 40-546 B.C.), Greek physical philosopher, son of Examyus and Cleobuline, is universally recog nized as the founder of Greek geometry, astronomy and philosophy.

00Of the fact that Thales visited Egypt, and there became acquainted with geometry, there is abundant evidence.

006 But the characteristic feature of the work of Thales was that to the knowledge thus acquired he added the capital creation of the geometry of lines, which was essentially abstract in its character.

00The only geometry known to the Egyptian priests was that of surfaces, together with a sketch of that of solids, a geometry consisting of some simple quadratures and elementary cubatures, which they had obtained empirically.

00other hand, introduced abstract geometry, the object of which is to establish precise relations between the different parts of a figure, so that some of them could be found by means of others in a manner strictly rigorous.

00The following discoveries in geometry are attributed to Thales (I) the circle is bisected by its diameter (Procl.

00Two applications of geometry to the solution of practical problems are also attributed to him: - (i) the determination of the distance of a ship at sea, for which he made use of the last theorem; (2) the determination of the height of a pyramid by means of the length of its shadow: according to Hieronymus of Rhodes (Diog.

00Proclus, too, in his summary of the history of geometry before Euclid, which he probably derived from Eudemus of Rhodes, says that Thales, having visited Egypt, first brought the knowledge of geometry into Greece, Assyrian Discoveries, p. 409.

00It is worth noticing that it was in this manner that this remarkable property of the circle, with which, in fact, abstract geometry was inaugurated, presented itself to the imagination of Dante: " 0 se del mezzo cerchio far si puote Triangol si, ch'un retto non avesse."

00In a scientific point of view: (a) we see, in the first place, that by his two theorems he founded the geometry of lines, which has ever since remained the principal part of geometry; (b) he may, in the second place, be fairly considered to have laid the foundation of algebra, for his first theorem establishes an equation in the true sense of the word, while the second institutes a proportion.'

00Lastly, in a practical point of view: Thales furnished the first example of an application of theoretical geometry to practice,' and laid the foundation of an important branch of the same - the measurement of heights and distances.

00For the further progress of geometry see Pythagoras.

00Allman, " Greek Geometry from Thales to Euclid," Hermathena, No.

00In 1886 he was chosen to succeed Felix Klein in the chair of geometry at Leipzig, but as his fame grew a special post was arranged for him in Christiania.

00A special application of his theory of continuous groups was to the general problem of non-Euclidean geometry.

00The latter part of the book above mentioned was devoted to a study of the foundations of geometry, considered from the standpoint of B.

00GASPARD MONGE (1746-1818), French mathematician, the inventor of descriptive geometry, was born at Beaune on the 10th of May 1746.

00And Monge, continuing his researches, arrived at that general method of the application of geometry to the arts of construction which is now called descriptive geometry (see Geometry, Descriptive).

00He at first intended to adopt the medical profession, and made some progress in anatomy, botany and chemistry, after which he studied chronology, geometry and astronomy.

00In July 1662 he was elected professor of geometry in Gresham College, on the recommendation of Dr John Wilkins, master of Trinity College and afterwards bishop of Chester; and in May 1663 he was chosen a fellow of the Royal Society, at the first election made by the council after obtaining their charter.

00(b) How then are the primary data of mathematical cognition to be derived from an experience containing space and time relations in Hume, in regard to this problem, distinctly separates geometry from algebra and arithmetic, i.e.

00With regard to geometry, he holds emphatically that it is an empirical doctrine, a science founded on observation of concrete facts.

00The rough appearances of physical facts, their outlines, surfaces and so on, are the data of observation, and only by a method of approximation do we gradually come near to such propositions as are laid down in pure geometry.

00He definitely repudiates a view often ascribed to him, and certainly advanced by many later empiricists, that the data of geometry are hypothetical.

00geometry) much excels, both in universality and exactness, the loose judgments of the senses and imagination, yet [it] never attains a perfect precision and exactness " (i.

00So far, then, as geometry is concerned, Hume's opinion is perfectly definite.

00Starting from this principle he was driven to geometry for insight into the ground and modes of emotion.

00Ward's colleague, the more famous John Wallis, Savilian professor of geometry from 1649, had been privy to the challenge thrown out in 1654, and it was arranged that they should critically dispose of the De corpore between them.

00The consequence was that, when not spending himself in vain attempts to solve the impossible problems that have always waylaid the fancy of self-sufficient beginners, he took an interest only in the elements of geometry, and never had any notion of the full scope of mathematical science, undergoing as it then was (and not least at the hands of Wallis) the extraordinary development which made it before the end of the century the potent instrument of physical discovery which it became in the hands of Newton.

00He was even unable, in dealing with the elementary conceptions of geometry, to work out with any consistency the few original thoughts he had, and thus became the easy sport of Wallis.

00With the translation,' in the spring of 1656, he had ready Six Lessons to the Professors of Mathematics, one of Geometry, the other of Astronomy, in the University of Oxford (E.W.

00181-356), in which, after reasserting his view of the principles of geometry in opposition to Euclid's, he proceeded to repel Wallis's objections with no lack of dialectical skill, and with an unreserve equal to Wallis's own.

00In its English form, as Seven Philosophical Problems and Two Propositions of Geometry (E.W.

00He was also the author of rhetorical exercises on hackneyed sophistical themes; of a Quadrivium (Arithmetic, Music, Geometry, Astronomy), valuable for the history of music and astronomy in the middle ages; a general sketch of Aristotelian philosophy; a paraphrase of the speeches and letters of Dionysius Areopagita; poems, including an autobiography; and a description of the Augusteum, the column erected by Justinian in the church of St Sophia to commemorate his victories over the Persians.

00grammar, music, painting, sculpture, medicine, geometry, mathematics and optics; c. 2 is on the general principles of architectural design; c. 3 on the considerations which determine a design, such as strength, utility, beauty; c. 4 on the nature of different sorts of ground for sites; c. 5 on walls of fortification; c. 6 on aspects towards the north, south and other points; c. 7 on the proper situations of temples dedicated to the various deities.

00The principal editions of Durer's theoretical writings are these :- Geometry and Perspective.

00He received £50 for a translation of Legendre's Geometry; and an introduction, explaining the theory of proportion, is said by De Morgan to show that he could have gained distinction as an expounder of mathematical principles.

00The 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.

00In the article Geometry, Analytical, it iS Shown that the general equation of the second degree represents a parabola when the highest terms form a perfect square.

00Analytic This is the analytical expression of the projective Geometry.

00See the bibliography to the articles Conic Sections; Geometry, Analytical; and Geometry, Projective.

00In the geometry of plane curves, the term parabola is often used to denote the curves given by the general equation a' n x n = ym+n, thus ax= y 2 is the quadratic or Apollonian parabola; a 2 x = y 3 is the cubic parabola, a 3 x = y4 is the biquadratic parabola; semi parabolas have the general equation ax n-1 = yn, thus ax e = y 3 is the semicubical parabola and ax 3 = y 4 the semibiquadratic parabola.

00Descartes used the curve to solve sextic equations by determining its intersections with a circle; mechanical constructions were given by Descartes (Geometry, lib.

00(See CURVE; and GEOMETRY, ANALYTICAL.)

00For the Euclidian axioms see Geometry.

00But he failed to obtain either of two posts - the professorships of mathematics at the Royal Military Academy and of geometry in Gresham College - for which he applied in 1854, though he was elected to the former in the following year on the death of his successful competitor.

00In 1883 he was chosen to succeed Henry Smith in the Savilian chair of geometry at Oxford, and there he produced his theory of reciprocants, largely by the aid of his "method of infinitesimal variation."

00SccSerca, twelve, and Spa, a face or base), in geometry, a solid enclosed by twelve plane faces.

00The geometry of sheet-metal work and of platers' and boiler-makers' work is identical up to a certain stage.

00On his return from a journey to Dalmatia, for the purpose of selecting and fortifying the port of Trieste, he was nominated, November 1703, Savilian professor of geometry at Oxford, and received an honorary degree of doctor of laws in 1710.

003 18 E) makes Protagoras pointedly refer to sophists who, " when young men have made their escape from the arts, plunge them once more into technical study, and teach them such subjects as arithmetic, astronomy, geometry and music."

00It may be imagined further that, when he established himself at the Academy, his first care was to draw up a scheme of education, including arithmetic, geometry (plane and solid), astronomy, harmonics and dialectic, and that it was not until he had arranged for the carrying out of this programme that he devoted himself to the special functions of professor of philosophy.

00arithmetic of number, geometry of magnitude, astronomy of stars, politics of government, ethics of goods.

00We need devices, indeed, to determine priority or superior claim to be " better known absolutely or in the order of nature," but on the whole the problem is fairly faced.4 Of science Aristotle takes for his examples sometimes celestial physics, more often geometry or arithmetic, sometimes a concrete science, e.g.

00Quaternions (as a mathematical method) is an extension, or improvement, of Cartesian geometry, in which the artifices of co-ordinate axes, &c., are got rid of, all directions in space being treated on precisely the same terms. It is therefore, except in some of its degraded forms, possessed of the perfect isotropy of Euclidian space.

00The evolution of quaternions belongs in part to each of two weighty branches of mathematical history - the interpretation of the imaginary (or impossible) quantity of common algebra, and the Cartesian application of algebra to geometry.

00To any one acquainted, even to a slight extent, with the elements of Cartesian geometry of three dimensions, a glance at the extremely suggestive constituents of this expression shows how justly Hamilton was entitled to say: " When the conception ...

00had been so far unfolded and fixed in my mind, I felt that the new instrument for applying calculation to geometry, for which I had so long sought, was now, at least in part, attained."

00There are many other systems, based on various principles, which have been given for application to geometry of directed lines, but those which deal with products of lines are all of such complexity as to be practically useless in application.

00Others, such as the Barycentrische Calciil of Mobius, and the Methode des equipollences of Bellavitis, give elegant modes of treating space problems, so long as we confine ourselves to projective geometry and matters of that order; but they are limited in their field, and therefore need not be discussed here.

00(1) Generation of the concept through imaginaries and development into a method applicable to Euclidean geometry.

00(See GEOMETRY, ï¿½ Analytical.)

00This is essentially a theorem of projective geometry, but the following statical proof is interesting.

00In the analytical investigations of line geometry, these six quantities, supposed subject to the relation (4), are used to specify a line, and are called the six co-ordinates of the line; they are of course equivalent to only four independent quantities.

00The modification of motion and the modification of force take place together, and are connected by certain laws; but in the study of the theory of machines, as well as in that of pure mechanics, much advantage has been gained in point of clearness and simplicity by first considering alone the principles of the modification of motion, which are founded upon what is now known as Kinematics, and afterwards considering the principles of the combined modification 01 motion and force, which are founded both on geometry and on the laws of dynamics.

00It is, indeed, the cardinal weakness of this form of intuitionism that no satisfactory list can be given and that no moral principles have the "constant and never-failing entity," or the definiteness, of the concepts of geometry.

00In Paris he witnessed the revolution of 1848, and only returned to Turin in 1852, when he taught applied geometry at the technical institute.

00Phys., 1905, 18, p. 941) founded his theory of aberrations on the differential geometry of surfaces.

00Monge in the chair of analytical geometry.

00His published mathematical works include: Analytic Geometry of Three Dimensions (1862), Treatise on Conic Sections (4th ed., 1863) and Treatise on the Higher Plane Curves (2nd ed., 1873); these books are of the highest value, and have been translated into several languages.

00In addition to this, he translated various other treatises, to the number, it is said, of sixty-six; among these were the Tables of "Arzakhel," or Al Zarkala of Toledo, Al Farabi On the Sciences (De scientiis), Euclid's Geometry, Al Farghani's Elements of Astronomy, and treatises on algebra, arithmetic and astrology.

00That he was, as stated by Archdeacon Thomas Martin, the author of a Life of Wykeham, published in 1597, taught classics, French and geometry by a learned Frenchman on the site of Winchester College, is a guess due to Wykeham's extant letters being in French and to the assumption that he was an architect.

00Philosophy, grammar, the history and theory of language, rhetoric, law, arithmetic, astronomy, geometry, mensuration, agriculture, naval tactics, were all represented.

00He studied mathematics, civil and military architecture, and astronomy, and became associate of the Academie des Sciences, professor of geometry, secretary to the Academy of Architecture and fellow of the Royal Society of London.

00He was much interested, too, in universal algebra, non-Euclidean geometry and elliptic functions, his papers "Preliminary Sketch of Bi-quaternions" (1873) and "On the Canonical Form and Dissection of a Riemann's Surface" (1877) ranking as classics.

00He also published several papers on algebraic forms and projective geometry.

00Among his most remarkable works may be mentioned his ten memoirs on quantics, commenced in 1854 and completed in 1878; his creation of the theory of matrices; his researches on the theory of groups; his memoir on abstract geometry, a subject which he created; his introduction into geometry of the "absolute"; his researches on the higher singularities of curves and surfaces; the classification of cubic curves; additions to the theories of rational transformation and correspondence; the theory of the twenty-seven lines that lie on a cubic surface; the theory of elliptic functions; the attraction of ellipsoids; the British Association Reports, 1857 and 1862, on recent progress in general and special theoretical dynamics, and on the secular acceleration of the moon's mean motion.

00Schering the Disquisitiones arithmeticae, (2) Theory of Numbers, (3) Analysis, (4) Geometry and Method of Least Squares, (5) Mathematical Physics, (6) Astronomy, and (7) the Theoria motus corporum coelestium.

00Eagles, Constructive Geometry of Plane Curves.

00Plucker aimed at furnishing modern geometry with suitable analytical methods so as to give it an independent analytical development.

00Steiner in cultivating geometry in its purely synthetic form.

00Induced by the encouragement of his mathematical friends in England, Plucker in 1865 returned to the field in which he first became famous, and adorned it by one more great achievement - the invention of what is now called "line geometry."

00He was engaged in bringing out a large work embodying the results of his researches in line geometry when he died on the 22nd of May 1868.

00Meanwhile he was filling his note-books as busily as ever with the results of his studies in statics and dynamics, in human anatomy, geometry and the phenomena of light and shade.

00In reply her correspondent says that the master is wholly taken up with geometry and very impatient of the brush, but at the same time tells her all about his just completed cartoon for the Annunziata.

00After receiving preliminary instructions in mathematics from his father, he was sent to the university of Basel, where geometry soon became his favourite study.

00Having taken his degree as master of arts in 1723, Euler applied himself, at his father's desire, to the study of theology and the Oriental languages with the view of entering the church, but, with his father's consent, he soon returned to geometry as his principal pursuit.

00He soon commenced to read the Principia, and at sixteen he had mastered a great part of that work, besides some more modern works on analytical geometry and the differential calculus.

00This post he held until 1898; but in 1892 he was also made professor of astronomy and geometry at Cambridge and director of the university observatory.

00His Elements of Geometry first appeared in 1795 and have passed through many editions; his Outlines of Natural Philosophy (2 vols., 1812-1816) consist of the propositions and formulae which were the basis of his class lectures.

00Playfair's contributions to pure mathematics were not considerable, his paper "On the Arithmetic of Impossible Quantities," that "On the Causes which affect the Accuracy of Barometrical Measurements," and his Elements of Geometry, all already referred to, being the most important.

00He therefore bought an English edition of Euclid with an index of propositions at the end of it, and, having turned to two or three which he thought likely to remove his difficulties, he found them so selfevident that he put aside Euclid " as a trifling book," and applied himself to the study of Descartes's Geometry.

00The study of Descartes's Geometry seems to have inspired Newton with a love of the subject, and to have introduced him to the higher mathematics.

00- By consulting an account of my expenses at Cambridge, in the years 1663 and 1664, I find that in the year 1664 a little before Christmas, I, being then Senior Sophister, bought Schooten's Miscellanies and Cartes' Geometry (having read this Geometry and Oughtred's Clavis clean over half a year before), and borrowed Wallis's works, and by consequence made these annotations out of Schooten and Wallis, in winter between the years 1664 and 1665.

00It was his duty as professor to lecture at least once a week in term time on some portion of geometry, arithmetic, astronomy, geography, optics, statics, or some other mathematical subject, and also for two hours in the week to allow an audience to any student who might come to consult with the professor on any difficulties he had met with.

00WITCH OF AGNESI, in geometry, a cubic curve invented by Maria Gaetana Agnesi.

00(See Geometry: Analytical.) Any relation whatever between (x, y) determines a curve, and conversely every curve whatever is determined by a relation between (x, y).

00The notion of imaginary intersections, thus presenting itself, through algebra, in geometry, must be accepted in geometry - and it in fact plays an all-important part in modern geometry.

00we state this generally without in the first instance, or it may be without ever, distinguishing whether these are real or imaginary; so in geometry we say that a curve of the mth order is met by an arbitrary line in m points, or rather we thus, through algebra, obtain the proper geometrical definition of a curve of the mth order, as a curve which is met by an arbitrary line in m points (that is, of course, in m, and not more than m,.

00The foregoing notion of a point at infinity is a very important one in modern geometry; and we have also to consider the paradoxical statement that in plane geometry, or say as regards the plane, infinity is a right line.

00This admits of an easy illustration in solid geometry.

001 1 In solid geometry infinity is a plane - its intersection with any given plane being the right line which iš the infinity of this given plane.

00The theorem is here referred to partly on account of its bearing on the theory of imaginaries in geometry.

00Bearing in a somewhat similar manner also on the theory of imaginaries in geometry (but the notion presents itself in a more explicit form), there is the memoir by L.

00There is in analytical geometry little occasion for any explicit use of line-co-ordinates; but the theory is very important; it serves to show that in demonstrating by point-co-ordinates any purely descriptive theorem whatever, we demonstrate the correlative theorem; that is, we do not demonstrate the one theorem, and then (as.

00In his admirable papers upon the modes of teaching arithmetic and geometry, originally published in the Quarterly Journal of Education (reprinted in The Schoolmaster, vol ii.), he remonstrated against the neglect of logical doctrine.

00In 1858 he became professor of mathematics at St Andrews, but lectured only for a session, when he vacated the chair for the Lowndean professorship of astronomy and geometry at Cambridge.

00In Chinese he published books on arithmetic, geometry, algebra (De Morgan's), mechanics, astronomy (Herschel's), and The Marine Steam Engine (T.

00The doctrine of geometrical continuity and the application of algebra to geometry, developed in the 16th and 17th centuries mainly by Kepler and Descartes, led to the discovery of many properties which gave to the notion of infinity, as a localized space conception, a predominant importance.

00A 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.

00These subjects are treated in Geometry.

00dimensio, a measuring), in geometry, a magnitude measured in a specified direction, i.e.

00The "fourth dimension" is a type of nonEuclidean geometry, in which it is conceived that a "solid" has one dimension more than the solids of experience.

00A determinant is symmetrical when every two elements symmetrically situated in regard to the dexter diagonal are equal to each other; if they are equal and opposite (that is, if the sum of the two elements be = o), this relation not extending to the diagonal elements themselves, which remain arbitrary, then the determinant is skew; but if the relation does extend to the diagonal terms (that is, if these are each = o), then the determinant is skew symmetrical; thus the determinants a, h, g a, v, - µ 0, v, - h, b, f - v, h, - v, 0, g,f,c c 12, - X, o are respectively symmetrical, skew and skew symmetrical: =0; a,b,c,d a' b' c' d'a" b c d" a, b, c, d a' b' c' d'a", b N' c N' dN,, , c d The theory admits of very extensive algebraic developments, and applications in algebraical geometry and other parts of mathematics.

00In astronomy, as in analytical geometry, the position of a point is defined by stating its distance and its direction from a point of reference taken as known.

00It must be admitted that any intelligent comprehension of the subject requires at least a grasp of the fundamental conceptions of analytical geometry and the infinitesimal calculus, such as only one with some training in these subjects can be expected to have.

00The invention of logarithms, the rise of analytical geometry, and the evolution of B.

00We hear also of one Master Peter, who inscribed and illuminated maps for the infante; the mathematician Pedro Nunes declares that the prince's mariners were well taught and provided with instruments and rules of astronomy and geometry "which all map-makers should know"; Cadamosto tells us that the Portuguese caravels in his day were the best sailing ships afloat; while, from several matters recorded by Henry's biographers, it is clear that he devoted great attention to the study of earlier charts and of any available information he could gain upon the trade-routes of north-west Africa.

00He also wrote or edited various Chinese works on geography, the celestial and terrestrial spheres, geometry and arithmetic. And the detailed history of the mission was drawn out by him, which after his death was brought home by P. Nicolas Trigault, and published at Augsburg, and later in a complete form at Lyons under the name De Expeditione Christiana apud Sinas Suscepta, ab Soc. Jesu, Ex P. Mat.

00The 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.

00In analytical geometry, r the equation axe+2hxy+bye+2gx+2fy+ c = o represents an ellipse when ab > h 2; if the centre of the curve be the origin, the equation is a 1 x 2 +2h 1 xy+b i y 2 =C 1, and if in addition a pair of conjugate diameters are the axes, the equation is further simplified to Ax e +By 2 = C. The simplest form is x 2 /a 2 +y 2 /b 2 = 1, in which the centre is the origin and the major and minor axes the axes of co-ordinates.

00iroXus, many, g 5pa, a base), in geometry, a solid figure contained by plane faces.

00These solids played an important part in the geometry of the Pythagoreans, and in their cosmology symbolized the five elements: fire (tetrahedron), air (octahedron), water (icosahedron), earth (cube), universe or ether (dodecahedron).

00A lesson in geometry, given by Ostilio Ricci to the pages of the grand-ducal court, chanced, tradition avers, to have Galileo for an unseen listener; his attention was riveted, his dormant genius was roused, and he threw all his energies into the new pursuit thus unexpectedly presented to him.

00More particularly, "minute" is used of the sixtieth part of any unit); in time, of an hour; and in astronomy, geometry, geography, &c., of a degree in the measurement of a circle.

00In geometry, and in geometrical crystallography, the term denotes a line which serves to aid the orientation of a figure.

00During his tenure of this chair he published two volumes of a Course of Mathematics - the first, entitled Elements of Geometry, Geometrical Analysis and Plane Trigonometry, in 1809, and the second, Geometry of Curve Lines, in 1813; the third volume, on Descriptive Geometry and the Theory of Solids was never completed.

00In ancient geometry the name was restricted to the three particular forms now designated the ellipse, parabola and hyperbola, and this sense is still retained in general works.

00But in modern geometry, especially in the analytical and projective methods, the "principle of continuity" renders advisable the inclusion of the other forms of the section of a cone, viz.

00In projective geometry it is convenient to define a conic section as the projection of a circle.

00In 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.

00In 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.

00The 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.

00Although 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.

00Johann Kepler (1571-1630) made many important discoveries in the geometry of conics.

00This subject is mathematically discussed in the article Geometry: Projective.

00While Desargues and Pascal were founding modern synthetic geometry, Rene Descartes was developing the algebraic representation of geometric relations.

00The 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.

00A mathematical investigation of the conics by this method is given in the article Geometry: Analytical.

00In 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.

00-For the ancient geometry of conic sections, especially of Apollonius, reference should be made to T.

00Michel 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.

00Eagles, 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.

00Reference may also be made to C. Taylor, An Introduction to Ancient and Modern Geometry of Conics (1881).

00See also list of works under GEOMETRY: Analytical and Projective.

00Jacobi and of the brothers Alexander and Wilhelm von Humboldt a new chair of geometry was founded for him at Berlin (1834).

00Steiner's mathematical work was confined to geometry.

00This he treated synthetically, to the total exclusion of analysis, which he hated, and he is said to have considered it a disgrace to synthetical geometry if equal or higher results were obtained by analytical methods.

00In his Systematische Entwickelung der Abhangigkeit geometrischer Gestalten von einander he laid the foundation of modern synthetic geometry.

00He showed that assigning meaning to the sign of an otherwise homogenous representation of geometry could provide a multitude of benefits.

00geometric algebra can be applied to any subject in mathematics, physics or engineering which is in some part rooted in geometry.

00algebraic geometry.

00analytic geometry is a such indicative case.

00Athens login off campus annals OF GLOBAL ANALYSIS AND GEOMETRY, from Swets 97- Access on and off campus.

00annulus geometry: design for rising line, constant mean diameter, falling line.

00assigning meaning to the sign of an otherwise homogenous representation of geometry could provide a multitude of benefits.

00axioms of Euclidean geometry, were considered to be self evident.

00The geometry of the polypeptide hydrogen bonds involving the peptide backbone atoms were also analyzed and shown to be fairly independent of sidechain influences.

00In facial recognition biometrics, the geometry of our faces is measured.

00The precise geometry and details of an X-ray producing region around a supermassive black hole are not known.

00boron hydrides is the twelve vertex B 12 H 12 2- anion that displays perfect icosahedral geometry.

00composite widgets here will deny any geometry request from their children by default.

00Numerical Geometry Ltd. Games Writing games is a good way of exploring strongly interactive computer graphics.

00Topic: conjugate Type: Keyword Use: Use conjugate gradients to optimize geometry, instead of second derivative based methods.

00coordination geometry of the titanium species.

00Radiometric corrections may be necessary due to variations in scene illumination and viewing geometry, atmospheric conditions, and sensor noise and response.

00deviated in the direction of roundness from the shape calculated on the basis of projective geometry.

00differentiable manifolds and the geometric structures which dominate Riemannian geometry.

00differential geometry, Higman received an MA.

00In this approach we use differential calculus and differential geometry both to filter and analyze multi-dimensional images.

00dimensional geometry.

00excoriated flesh, frustrated sex, the geometry of fear.

00fractal geometry to evolutionary progress, digital technologies do simulate aspects of the natural world.

00Racing geometry and a lightweight aluminum frame are the key to its great handling.

00framing rectangle is used by the subclasses of RectangularShape to define their geometry.

00geometry of spacetime principally resides on the left-hand side, this situation seems unsatisfactory.

00Main arguments: current geometry, search direction, step, current gradient norm; on exit: optimized geometry, gradient norm.

00What about physical space - is Euclidean geometry really true for all space?

00From the early 1800s Gauss had an interest in the question of the possible existence of a non-Euclidean geometry.

00Using the techniques of fractal geometry, which he himself invented, Mandelbrot believes he has finally realized his ambition.

00Professor Griffiths is well known for his work in algebraic geometry.

00Even limited space in which we move they might not be true - projective geometry.

00To understand the course the student should know the basic ideas of Riemannian geometry.

00geometry optimization.

00geometry optimization was used to find a low energy (stable) shape.

00geometry mismatch.

00geometry theorem.

00Hence, everything in the physical world was a manifestation of spacetime geometry.

00For wings with full-span leading-edge devices a factor, dependent on planform geometry, is applied to allow for three-dimensional effects.

00Please see the new steering geometry page for an explanation of why we have done this.

00In a four-wheeled car rear suspension geometry can be designed to help point a car into a corner.

00Apart from engine set-up, Hofmann also tested a revised rear swinging arm and chassis geometry.

00The thermal and electrical field distribution in the waveguide of any electrode geometry can be determined.

00The upper limit for the binary boron hydrides is the twelve vertex B 12 H 12 2- anion that displays perfect icosahedral geometry.

00hyperbolic geometry on the flat plane.

00inaugural lecture on 12 October 1909 on ' The nature of geometry ' in which he outlined his research program.

00All the new bi-wire cables in the Original Range feature an enhanced geometry that minimizes cable inductance by precisely spacing opposing sets of conductors.

00infinitesimal geometry were equivalent to those used by the ancient Greeks.

00The geometry of the s-trans isomer also shows a significant double bond character for the N-P bond.

00iterations of that geometry that are already in process.

00Changes in magnetic field geometry while the spacecraft is traversing the transition region between the outer and middle magnetosphere are also considered.

00It was studied by ancient mathematicians due to its frequent appearance in geometry.

00misfireber of teams had a day to forget: Autocare retired early with broken steering geometry and Andy Racing with a misfiring engine.

00Typically a media failure or by a geometry mismatch.

00The monomer library files describe the internal geometry of a monomer library files describe the internal geometry of a monomer - they may contain complete or minimal descriptions of the monomers.

00The neutron fluence rate depends upon factors such as target thickness, charged particle beam current, geometry and required neutron fluence rate depends upon factors such as target thickness, charged particle beam current, geometry and required neutron energy resolution.

00non-Euclidean geometry.

00PainlevÃ© equations arise in many applications including models of water waves, nonlinear optics, general relativity, and in differential geometry.

00optimized geometry is printed here.

00Note 16: The fully optimized geometry is printed here.

00fiber aspect ratio and fiber orientation distribution in molded components, influence of mold geometry and molding conditions.

00photoelectron diffraction is a method used to determine the geometry of molecules or atoms on single crystal surfaces.

00planar geometry.

00pockmark formation, their geometry, age and distribution, and the sources of gas in the underlying geological strata are discussed.

00Instead he studied geometry in which the fifth postulate does not necessarily hold.

00projective geometry.

00These additional color constraints result in more accurately reconstructed geometry, which projects to better synthesized virtual views of the scene.

00redshift surveys are being undertaken to gain an understanding of the geometry of the Universe in three spatial dimensions.

00The most significant discoveries of KSU include non-Euclidean geometry, obtaining aniline from nitrobenzene, new element ruthenium, phenomenon of EPR.

00sacred geometry to be found in these cultures, Peter's painting provides an image for thought.

00Women's version of Specialized's super comfy Body Geometry saddle.

00The final three books are on solid geometry, and conclude with the construction and classification of the five Platonic solids.

00solid geometry of icosahedra site has been re-opened and is well worth a visit.

00The excellent and comprehensive solid geometry of icosahedra site has been re-opened and is well worth a visit.

00The grid symbolizing the roots is open, and contrasts both with the apparent solidity of the sun and with its differing geometry.

00Abu'l-Wafa and Abu Nasr Mansur both applied spherical geometry to astronomy and also used formulas involving sin and tan.

00This group studies the convection regime within the Earths Mantle in three dimensional spherical geometry at approaching Earth-like vigor.

00spoiler --- Top of page Sections: Pure, Games, Geometry, Physics, Large.

00These sketches are intended to show geometry of roofs at various spans and pitches and not structural details.

00If a crop geometry is specified a subregion of the image is obtained.

00tetrahedral geometry.

00thematic maps can only be created on maps with a region geometry.

00three-dimensional geometry.

00However, solutions like black holes, have a Euclidean geometry with non trivial topology.

00toric geometry, derived categories, quiver representations and the McKay correspondence.

00The accuracy of the LBL in deepwater is a result of the geometry of the seabed transponder array.

00treatise on universal science that included a 100-page appendix on geometry, containing his fundamental contributions to analytic geometry.

00triathlon bike usually has a steeper geometry with a STA greater than 76° .

00trigonal pyramidal geometry predicted by VSEPR theory.

00Continue geometry, including Pythagoras ' Theorem and basic trigonometry.

00Those 166 horses gallop along courtesy of a variable geometry turbocharger.

00The square air ducts ahead of each engine feature variable geometry vanes.

00vapourdesktop pattern has the feel of a vapor trail crossed with the geometry of mucus strands.

00Many of the composite widgets here will deny any geometry request from their children by default.

00- The range and importance of the scientific labours of Archimedes will be best understood from a brief account of those writings which have come down to us; and it need only be added that his greatest work was in geometry, where he so extended the method of exhaustion as originated by Eudoxus, and followed by Euclid, that it became in his hands, though purely geometrical in form, actually equivalent in several cases to integration, as expounded in the first chapters of our text-books on the integral calculus.

00(9) A Collection of Lemmas, consisting of fifteen propositions in plane geometry.

00TErpa-, four, Spa, face or base), in geometry, a solid bounded by four triangular faces.

00That idea of a method grew up with his study of geometry and arithmetic, - the only branches of knowledge which he would allow to be " made sciences."

004 The mathematics of which he thus speaks included the geometry of the ancients, as it had been handed down to the modern world, and arithmetic with the developments it had received in the direction of algebra.

00The arithmetical half of mathematics, which had been gradually growing into algebra, and had decidedly established itself as such in the Ad logisticen speciosam notae priores of Francois Vieta (1540-1603), supplied to some extent the means of generalizing geometry.

00Hence the world is left open for the free play of mechanics and geometry.

00But the pupil soon found his teacher to be but a charlatan, and betook himself, aided by commentaries, to master logic, geometry and the Almagest.

00(See are GEOMETRY: Analytical; Projective.)

00Napier deliberately set himself to abbreviate multiplications and divisions - operations of so fundamental a character that it might well have been thought that they were in rerum natura incapable of abbreviation; and he succeeded in devising, by the help of arithmetic and geometry alone, the one 1 The title runs as follows: Arithmetica Logarithmica, sive Logarithmorum chiliades triginta....

00After publishing a few short papers relating to theory of numbers and to geometry, he devoted himself to a thorough examination of the writings of K.

00We see from a statement of Cassiodorus that he furnished manuals for the quadrivium of the schools of the middle ages (the " quattuor matheseos disciplinae," as Boetius calls them) on arithmetic, music, geometry and astronomy.

00Government, and was a " conscientious objector," throughout the World War, was fined boo and £10 costs for making statements calculated to prejudice recruiting, and, in consequence, Trinity College, Cambridge, deprived him of his lectureship. His chief published works, on which his philosophical reputation was based up to the outbreak of the World War, were German Social Democracy (1896); Essay on the Foundations of Geometry (1897); Principles of Mathematics (5903); Principia Mathematica (with A.

00In its charter this institution is described as "an academy for the purpose of promoting piety and virtue, and for the education of youth in the English, Latin and Greek languages, in writing, arithmetic, music and the art of speaking, practical geometry, logic and geography, and such other of the liberal arts and sciences or languages, as opportunity may hereafter permit."

001865-1866), is occupied with the investigation of the projective properties of figures (see Geometry).

00In geometry, a bipartite curve consists of two distinct branches (see Parabola, figs.

00Furthermore, can we not complete the circle of the mathematical sciences by adding geometry ?

00It is sufficient to observe here that they are concerned with special types of classes of classes and of classes of relations, and that the connexion of geometry with number and magnitude is in no way an essential part of the foundation of the science.

00He showed at an early age wellmarked mathematical powers, and his progress was so rapid in arithmetic and geometry that he was soon beyond the guidance not only of his father but of schoolmaster Schulz, who assisted in the mathematical department of his training.

00Of the customary three themes which he suggested for his trial lecture, that "On the Hypotheses which form the Foundation of Geometry" was chosen at the instance of Gauss, who was curious to hear what so young a man had to say on this difficult subject, on which he himself had in private speculated so pro foundly (see Geometry, Non-Euclidian).

00He published in 15 books a treatise on the New Geometry (1587), and works on history, rhetoric and the art of war.

00This reaction has taken the form of a return to the alliance between algebra and geometry (Ã¯¿½5), on which modern analytical geometry is based; the alliance, however, being concerned with the application of graphical methods to particular cases rather than to general expressions.

00These applications are sometimes treated under arithmetic, sometimes under algebra; but it is more convenient to regard graphics as a separate subject, closely allied to arithmetic, algebra, mensuration and analytical geometry.

00The association of algebra with arithmetic on the one hand, and with geometry on the other, presents difficulties, in that geometrical measurement is based essentially on the idea of continuity, while arithmetical measurement is based essentially on the idea of discontinuity; both ideas being equally matters of intuition.

00In his famous Geometria (1637), which is really a treatise on the algebraic representation of geometric theorems, he founded the modern theory of analytical geometry (see Geometry), and at the same time he rendered signal service to algebra, more especially in the theory of equations.

00== Treatise on the Differential Calculus and the Elements of the Integral Calculus (1852, 6th ed., 1873), Treatise on Analytical Statics (1853, 4th ed., 1874); Treatise on the Integral Calculus (1857, 4th ed., 1874); Treatise on Algebra (1858, 6th ed., 1871); Treatise on Plane Coordinate Geometry (1858, 3rd ed., 1861); Plane Trigonometry (1859, 4th ed., 1869); Spherical Trigonometry (1859); History of the Calculus of Variations (1861); Theory of Equations (1861, 2nd ed.

00acf aZpa, a ball or globe), in geometry, the solid or surface traced out by the revolution of a semicircle about its diameter; this is essentially Euclid's definition; 1 in the modern geometry of surfaces it is defined as the quadric surface passing through the circle at infinity.

00In this article the equations to the more important circles - the circumscribed, inscribed, escribed, self-conjugate--will be given; reference should be made to the article Triangle for the consideration of other circles (nine-point, Brocard, Lemoine, &c.); while in the article Geometry: Analytical, the principles of the different systems are discussed.

00+214'(37+2V 'ya+2w'a(3=o, it is readily seen that for by the relation Ea 2 (p - q) (p - r) = 4 0 2 (see Geometry: Analytical, which is generally written lap, bq, cr} 2 = 4 z.

00Rectification and quadrature of the circle have thus been, since the time of Archimedes at least, practically identical problems. Again, since the circumferences of circles are proportional to their diameters - a proposition assumed to be true from the dawn almost of practical geometry - the rectification of the circle is seen to be transformable into finding the ratio of the circumference to the diameter.

00Euclid (Elements, book 1) defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other (see Geometry, Euclidean).

00He took a very active part in the measures for the establishment of the normal school (which existed only during the first four months of the year 1795), and of the school for public works, .afterwards the polytechnic school, and was at each of them professor for descriptive geometry; his methods in that science were first published in the form in which the shorthand writers took down his lessons given at the normal school in 1795, and .again in 1798-1799.

00Hence the title of his new piece: Ertl/mat eyewperpias, aypoudas, avruroXcreias, apaBeias, or Marks of the Absurd Geometry, Rural Language, Scottish Church Politics, and Barbarisms of John Wallis, Professor of Geometry and Doctor of Divinity (E.

00The preface treats of Greek sciences, geometry, the discovery of specific gravity by Archimedes, and other discoveries of the Greeks, and of Romans of his time who have vied with the Greeks -- Lucretius in his poem De Rerum Natura, Cicero in rhetoric, and Varro in philology, as shown by his De Lingua Latina.

00But no religious paintings on the grand scale, corresponding to these drawings of 1521-1524, were ever carried out; perhaps partly because of the declining state of the artist's health, but more because of the degree to which he allowed his time and thoughts to be absorbed in the preparation of his theoretical works on geometry and perspective, proportion and fortification.

00To the tale of his woodcuts, besides a few illustrations to his book on measurements (that is, geometry and perspective), and on fortification, he only added one Holy Family and one portrait, that of his friend Eoban Hesse.

00Of his theoretical books, he only succeeded in getting two finished and produced during his lifetime, that on geometry and perspective or measurement, to use his own title - which was published at Nuremberg in 1525, and that on fortification, published in 1527; the work on human proportions was brought out shortly after his death in 1528.

00A fundamental property of the curve is that the line at infinity is a tangent (see Geometry, Projective), and it follows that the centre and the second real focus and directrix are at infinity.

00The equations to the tangent and normal at the point x' y are yy' = 2a(x+x) and aa(y - y')+y'(x - x')=o, and may be obtained by general methods (see Geometry, Analytical, and Infinitesimal Calculus).

00The scope of his researches was described by Arthur Cayley, his friend and fellow worker, in the following words: "They relate chiefly to finite analysis, and cover by their subjects a large part of it - algebra, determinants, elimination, the theory of equations, partitions, tactic, the theory of forms, matrices, reciprocants, the Hamiltonian numbers, &c.; analytical and pure geometry occupy a less prominent position; and mechanics, optics and astronomy are not absent."

00(See GEOMETRY, Ã¯¿½ Analytical.)

00KuKAos, circle, and Ethos, form), in geometry, the curve traced out by a point carried on a circle which rolls along a straight line.

00In the first volume of this treatise Plucker introduced for the first time the method of abridged notation which has become one of the characteristic features of modern analytical geometry (see Geometry, Analytical).

00The work was so far advanced that his pupil and assistant Felix Klein was able to complete and publish it (see Geometry, Line).

00The former treatise is historically interesting for the light it throws on the development which the geometry of the sphere had already reached even before Autolycus and Euclid (see THEODOSrus OF Tripolis).

00Trilinear co-ordinates (see Geometry: Analytical) were first used by E.

00(See Geometry: Analytical.) The Cartesian co-ordinates (x, y) and trilinear co-ordinates (x, y, z) are point-co-ordinates for determining the position of a point; the new co-ordinates, say (, 77, 0 are line-co-ordinates for determining the position of a line.

00Mensuration of the Platonic Solids.-The mensuration of the regular polyhedra is readily investigated by the methods of elementary geometry, the property that these solids may be inscribed in and circumscribed to concentric spheres being especially useful.

00Kepler's greatest contribution to geometry lies in his formulation of the "principle of continuity" which enabled him to show that a parabola has a "caecus (or blind) focus" at infinity, and that all lines through this focus are parallel (see Geometrical Continuity).

00This assumption (which differentiates ancient from modern geometry) has been developed into one of the most potent methods of geometrical investigation (see Geometry: Projective).

00In particular, the study of toric geometry, derived categories, quiver representations and the McKay correspondence.

00Many redshift surveys are being undertaken to gain an understanding of the geometry of the Universe in three spatial dimensions.

00We also have a Black Glass standard (0% Reflectance in 0/45 Geometry).

00The most significant discoveries of KSU include non-Euclidean geometry, obtaining aniline from nitrobenzene, new element Ruthenium, phenomenon of EPR.

00Like the sacred geometry to be found in these cultures, Peter 's painting provides an image for thought.

00Women 's version of Specialized 's super comfy Body Geometry saddle.

00Spoiler --- Top of page Sections: Pure, Games, Geometry, Physics, Large.

00The geometry of the ZT 's MacPherson strut front suspension has been tuned to the requirements of the new chassis platform.

00Ice is usually composed of a lattice of water molecules arranged with perfect tetrahedral geometry.

00Note: D ot density thematic maps can only be created on maps with a region geometry.

00The final three books deal with aspects of three-dimensional geometry.

00This was a philosophical treatise on universal science that included a 100-page appendix on geometry, containing his fundamental contributions to analytic geometry.

00A triathlon bike usually has a steeper geometry with a STA greater than 76°.

00This is in conflict with the trigonal pyramidal geometry predicted by VSEPR theory.

00A cross between a middle school geometry lesson and a tub of lard in meltdown.

00The burners have variable geometry combustion heads, allowing for greater turndown capability and flexibility in installing the burners on different applications.

00The desktop pattern has the feel of a vapor trail crossed with the geometry of mucus strands.

00Geometry problems, white streaks and washed-out images are symptomatic of this.

00If this is the case, chances are it will look almost like a high school geometry assignment with grids, lines, dots and angles.

00It is a game that shows, in practice, the principles of geometry and solid body physics.

00One tab is for trigonometry and geometry functions, the second tab is for higher math functions and the third tab is for numbers you place in memory.

00All fields of math are represented, from algebra to chemistry to geometry and from finance to fractions to statistics.

00All it takes is some basic geometry lessons and a couple other tricks, and you'll have the best family portraits on the block.

00Discover ways to bond with others over a shared interest, whether it's your favorite band or your hatred of geometry.

00Are you having trouble with your latest geometry problem?

00You perhaps already know and may be surprised to learn that you've even used the Golden Ratio in geometry.

00It is only possible to derive Phi using mathematics, geometry or the Fibonacci Sequence of numbers.

00You can only derive Phi by using mathematics, geometry or the Fibonacci Sequence of numbers.

00Use real street signs to teach lessons on shapes and Geometry.

00Teaching Textbooks CD-ROM courses for Algebra, Geometry and Pre-Calculus.

00For older children who may be struggling to learn geometry, this type of hands on activity can be fun and educational.

00As part of a mathematics project, the creation of an origami frog was integrated into a geometry lesson.

00But how do you wear it without looking like your ninth grade geometry teacher or in the very least, the stuffiest version of yourself possible?

00Shapes - Younger children will want to start with squares, circles, triangles, rectangles and ovals but older children could use more advanced geometric shapes to reinforce their geometry lessons.

00Battleship coordinates game geometry helps players determine where plots are on a plane so that they can figure out where their opponents' ships are.

00As you play, you can use basic math or even geometry.

00While there is some luck associated with Battleship, there are some Battleship coordinates game geometry strategy that you can use in order to sink your opponent's battleship efficiently and quickly.

00This is the best way to use Battleship coordinate game geometry, but you can narrow down where ships are in a quick way.

00Using Battleship coordinates game geometry can work for you or against you.

00By using the Battleship coordinates game geometry to setup your ships and to determine where your opponent's ships are, you can make the game more strategic than lucky.

00Standard black and white offers a geometry that is both pleasing to the eye and to your design.

00Because the carat is a measure of weight rather than size, two stones that have identical carat weights may actually appear very different depending on their exact geometry, proportions, and shape.

00Geometry: The geometry of a princess cut must be precise.

00Because many black pearl ring designs incorporate multiple diamonds, a good understanding of the importance of diamond geometry and total carat weight are also valuable.

00Diamond Geometry: The precise geometry of the diamond, regardless of its overall shape, is critical.

00The art deco era is known for bold geometry and colorful styles, but heart shapes are not typically found in art deco engagement rings.

00Faux Hearts: Using more basic diamond shapes to create a simulated heart emphasizes the beautiful geometry of an art deco style.

00Prongs: Modern jewelers are able to precisely fashion the tips of prongs into miniscule hearts that are a fun accent to an art deco ring design without overwhelming the ring's geometry.

00Edwardian Engagement Rings: Edwardian rings are just as elaborate as art deco styles but incorporate more curves and natural shapes instead of angular geometry.

00Art Deco(1915-1938):Geometry, symmetry, and bold design.

00Puzzle games foster skills such as hand-eye coordination, fine motor skills, problem-solving ability, logic and reasoning, and fundamentals of geometry and spatial awareness.

00Learning shapes is an introduction to geometry.

00Math Playground has several fun math games that teach basic math, algebra, geometry, money and percentages.

00Board games: Educational Learning Games carries a large selection of games designed to teach children math facts, whether it's simple geometry through the use of puzzles, word problems or decimals.

00

The word usage examples above have been gathered from various sources to reflect current and historial usage. They do not represent the opinions of YourDictionary.com.