# Geometrical Sentence Examples

The windows in the outer walls are filled with pierced stone screens of

**geometrical**design.This, and the inclination of the orbit being given, we have all the

**geometrical**data necessary to compute the coordinates of the planet itself.The "axioms" of geometry are the fixed conditions which occur in the hypotheses of the

**geometrical**propositions.By

**geometrical**consideration it can be shown that the angle subtended by p, as seen from F, must be inversely as the square of its distance r.For the subjects under this heading see the articles CONIC SECTIONS; CIRCLE; CURVE;

**GEOMETRICAL**CONTINUITY; GEOMETRY, Axioms of; GEOMETRY, Euclidean; GEOMETRY, Projective; GEOMETRY, Analytical; GEOMETRY, Line; KNOTS, MATHEMATICAL THEORY OF; MENSURATION; MODELS; PROJECTION; Surface; Trigonometry.Reference to a

**geometrical**interpretation seems at first sight to throw light on the meaning of a differential coefficient; but closer analysis reveals new difficulties, due to the**geometrical**interpretation itself.The harmony between arithmetical and

**geometrical**measurement, which was disturbed by the Greek geometers on the discovery of irrational numbers, is restored by an unlimited supply of the causes of disturbance.The

**geometrical**theory, which formed the basis of the investigations of Descartes and Newton, afforded no explanation of the supernumerary bows, and about a century elapsed before an explanation was forthcoming.The

**geometrical**theory first requires a consideration of the path of a ray of light falling upon a transparent sphere.The

**geometrical**theory can afford no explanation of these coloured bands, and it has been shown that the complete phenomenon of the rainbow is to be sought for in the conceptions of the wave theory of light.AdvertisementThe mathematical discussion of Airy showed that the primary rainbow is not situated directly on the line of minimum deviation, but at a slightly greater value; this means that the true angular radius of the bow is a little less than that derived from the

**geometrical**theory.The borders of the garment are painted with

**geometrical**patterns in vivid colours; a broad stripe of ornament runs down the centre of the skirt.'Akin to the

**geometrical**works is that On the Dioptra, a remarkable book on land-surveying, so called from the instrument described in it, which was used for the same purposes as the modern theodolite.Many derivatives are known, some of which exist in two structural forms, exhibiting

**geometrical**isomerism after the mode of fumaric and maleic acids.Neither by

**geometrical**, nor physical, nor metaphysical principles had he succeeded in reaching and grasping the infinite and the spiritual, or in elucidating their relation to man and man's organism, though he had caught glimpses of facts and methods which he thought only required confirmation and development.AdvertisementThis capacity is then a function of the

**geometrical**dimensions of the conductor, and can be mathematically determined in certain cases.At the conclusion of his philosophical studies at the university, some

**geometrical**figures, which fell in his way, excited in him a passion for mathematical pursuits, and in spite of the opposition of his father, who wished him to be a clergyman, he applied himself in secret to his favourite science.He studied medicine and became a physician, but his attention was early directed also to

**geometrical**studies.None of them, in point of fact, has held its ground, and even his proposal to denote unknown quantities by the vowels A, E, I, 0, u, Y - the consonants B, c, &c., being reserved for general known quantities - has not been taken up. In this denotation he followed, perhaps, some older contemporaries, as Ramus, who designated the points in

**geometrical**figures by vowels, making use of consonants, R, S, T, &c., only when these were exhausted.The attempt to apply numerical methods to the comparison of

**geometrical**quantities led to the doctrine of incommensurables, and to that of the infinite divisibility of space.AdvertisementThey correspond to the two methods of regarding quantity - the arithmetical and the

**geometrical**.In its

**geometrical**aspect it is a mere**geometrical**point.His Hamburg interests continued from that date onwards to multiply in something like

**geometrical**progression.Up to a certain point, formulae of practical importance can be obtained by the use of elementary arithmetical or

**geometrical**methods.Mensuration involves the use of

**geometrical**theorems, but it is not concerned with problems of**geometrical**construction.AdvertisementOn the other hand, it is worth noticing that the words " quadrature " and " cubature " are originally due to

**geometrical**rather than numerical considerations; the former implying the construction of a square whose area shall be equal to that of a given surface, and the latter the construction of a cube whose volume shall be equal to that of a given solid.As a result of the importance both of the formulae obtained by elementary methods and of those which have involved the previous use of analysis, there is a tendency to dissociate the former, like the latter, from the methods by which they have been obtained, and to regard mensuration as consisting of those mathematical formulae which are concerned with the measurement of

**geometrical**magnitudes (including lengths), or, in a slightly wider sense, as being the art of applying these formulae to specific cases.This 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.It is in reference to the measurement of areas and volumes that it is of special importance to illustrate

**geometrical**truths by means of concrete cases.This method is unconsciously adopted by the teacher who illustrates the equality of area of two

**geometrical**figures by cutting them out of cardboard of uniform thickness and weighing them.The next stage is

**geometrical**mensuration, where**geometrical**methods are applied to determine the areas of plane rectilinear figures and the volumes of solids with plane faces.The first step is the establishment of the exact equality of congruence of two

**geometrical**figures.In the case of solid figures a more difficult

**geometrical**abstraction is involved.A collection of formulae relating to the circle, for instance, would comprise not only

**geometrical**and trigonometrical formulae, but also approximate formulae, such as Huygens's rule (§ 91), which are the result of advanced analysis.The ideas of moment and of centroid are extended to

**geometrical**figures, whether solid, superficial or linear.The mensuration of the cube, and its relations to other

**geometrical**solids are treated in the article Polyhedron; in the same article are treated the Archimedean solids, the truncated and snubcube; reference should be made to the article Crystallography for its significance as a crystal form.All these solutions were condemned by Plato on the ground that they were mechanical and not

**geometrical**, i.e.However, no proper

**geometrical**solution, in Plato's sense, was obtained; in fact it is now generally agreed that, with such a restriction, the problem is insoluble.In 1714 Ditton published his Discourse on the Resurrection of Jesus Christ; and The New Law of Fluids, or a Discourse concerning the Ascent of Liquids in exact

**Geometrical**Figures, between two nearly contiguous Surfaces.It is curious, however, to find that an ancient nation of the East, so wise in

**geometrical**proportions, should have followed what by modern experience may be regarded as an inverse method, that of obtaining a unit of length by deducing it through weights and cubic measure, rather than by deriving cubic measure through the unit of length.The table extends to 2 3 0270 in the arithmetical column, and it is shown that 230270.022 corresponds to 9.9999 9999 or 109 in the

**geometrical**column; this last result showing that (1.0001)23027 022 = 10.But the fantastic relations imagined by him of planetary movements and distances to musical intervals and

**geometrical**constructions seemed to himself discoveries no less admirable than the achievements which have secured his lasting fame.This problem, also termed the " Apollonian problem," was demonstrated with the aid of conic sections by Apollonius in his book on Contacts or Tangencies;

**geometrical**solutions involving the conic sections were also given by Adrianus Romanus, Vieta, Newton and others.The 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.This correlative numerical problem and the two purely

**geometrical**problems are inseparably connected historically.His achievement was a closely approximate D

**geometrical**solution of the FIG.For a few approximate

**geometrical**solutions, see Leybourn's Math.Restorations have been given by Marino Ghetaldi, by Hugo d'Omerique (

**Geometrical**Analysis, Cadiz, 1698), and (the best) by Samuel Horsley (1770).Lord Rayleigh's expression for the resolving power of different instruments is based on the assumption that the

**geometrical**image of the slit is narrow compared with the width of the diffraction image.The three commonest means are the arithmetical,

**geometrical**, and harmonic; of less importance are the contraharmonical, arithmetico-**geometrical**, and quadratic.The

**geometrical**mean of n quantities is the nth root of their product.The arithmetico-

**geometrical**mean of two quantities is obtained by first forming the**geometrical**and arithmetical means, then forming the means of these means, and repeating the process until the numbers become equal.He generalized Weber's law in the form that sensation generally increases in intensity as the stimulus increases by a constant function of the previous stimulus; or increases in an arithmetical progression as the stimulus increases in a

**geometrical**ratio; or increases by addition of the same amount as the stimulus increases by the same multiple; or increases as the logarithm of the stimulus.The

**geometrical**axis of the magnet is sometimes defined by means of a mirror rigidly attached to the magnet and having the normal to the mirror as nearly as may be parallel to the magnetic axis.This arrangement is not very convenient, as it is difficult to protect the mirror from accidental displacement, so that the angle between the

**geometrical**and magnetic axes may vary.In this case the

**geometrical**axis is the line joining the central division of the scale to the optical centre of the lens.Its characteristics were a flamboyant and fantastic treatment of plant and animal (though not of human) forms, a free use of the

**geometrical**device called the " returning spiral," and much skill in enamelling.The urns themselves are of clay, somewhat badly baked, and bear

**geometrical**patterns applied with a punch.The volume contains also dissertations on Logarithms and on the Limits of Quantities and Ratios, and a few problems illustrative of the ancient

**geometrical**analysis.Lie was a foreign member of the Royal Society, as well as an honorary member of the Cambridge Philosophical Society and the London Mathematical Society, and his

**geometrical**inquiries gained him the muchcoveted honour of the Lobatchewsky prize.It is obvious that at the end of n such operations the charge on A will be r n Q, so that the charge goes on increasing in

**geometrical**progression.He was undoubtedly a clear-sighted and able mathematician, who handled admirably the severe

**geometrical**method, and who in his Method of Tangents approximated to the course of reasoning by which Newton was afterwards led to the doctrine of ultimate ratios; but his substantial contributions to the science are of no great importance, and his lectures upon elementary principles do not throw much light on the difficulties surrounding the border-land between mathematics and philosophy.If taken in isolation this passage might appear sufficient justification for Kant's view that, according to Hume,

**geometrical**judgments are analytical and therefore perfect.But it is to be recollected that, according to Hume, an idea is actually a representation or individual picture, not a notion or even a schema, and that he never claims to be able to extract the predicate of a

**geometrical**judgment by analysis of the subject.The properties of this individual subject, the idea of the triangle, are, according to him, discovered by observation, and as observation, whether actual or ideal, never presents us with more than the rough or general appearances of

**geometrical**quantities, the relations so discovered have only approximate exactness.Any exactitude attaching to the conclusions of

**geometrical**reasoning arises from the comparative simplicity of the data for the primary judgments.If we were to say that on his view the essential step must be the establishment of identities or equivalences, we should probably be doing justice to his doctrine of numerical reasoning, but should have some difficulty in showing the application of the method to

**geometrical**reasoning.For in the latter case we possess, according to Hume, no standard of equivalence other than that supplied by immediate observation, and consequently transition from one premise to another by way of reasoning must be, in

**geometrical**matters, a purely verbal process.The first desideratum here mentioned - the want, namely, of an accurate statement of the relation between the increase of population and food - Malthus doubtless supposed to have been supplied by the celebrated proposition that "population increases in a

**geometrical**, food in an arithmetical ratio."In 1654 Seth Ward (1617-1689), the Savilian professor of astronomy, replying in his Vindiciae academiarum to some other assaults (especially against John Webster's Examen of Academies) on the academic system, retorted upon Hobbes that, so far from the universities being now what he had known them in his youth, he would find his

**geometrical**pieces, when they appeared, better understood there than he should like.The simple plain ashlar masonry still predominates, but the wall surface is broken up with sunk panels, sometimes with

**geometrical**patterns in them.In the "Adam and Eve" of the next year, we find Diirer treating the human form in an entirely opposite manner; constructing it, that is, on principles of abstract

**geometrical**proportion.The decoration consists, as a rule, of stiff, conventional foliage, Arabic inscriptions, and

**geometrical**patterns wrought into arabesques of almost incredible intricacy and ingenuity.They are lined on the inside with upright slabs, on which are painted

**geometrical**figures and representations of animals.In addition to publishing a number of works on

**geometrical**and mechanical subjects, Poinsot also contributed a number of papers on pure and applied mathematics to Lionville's Journal and other scientific periodicals.That which is inscribed with the name of "Midas the King" is the most remarkable example of one class, in which a large perpendicular surface of rock is covered with a

**geometrical**pattern of squares, crosses and maeanders, surmounted by a pediment supported in the centre by a pilaster in low relief.The heraldic type is used on the monuments which appear to be the older, and the

**geometrical**pattern is often employed on the inscribed monuments, which are obviously later than the earliest uninscribed.Carpets with

**geometrical**patterns of the Midas-tomb style are occasionally found at the present time in the houses of the peasantry of the district.Their cloth is generally ornamented with

**geometrical**patterns.In crystallography, the regular or ordinary dodecahedron is an impossible form since the faces cut the axes in irrational ratios; the "pentagonal dodecahedron" of crystallographers has irregular pentagons for faces, while the

**geometrical**solid, on the other hand, has regular ones.The "rhombic dodecahedron," one of the

**geometrical**semiregular solids, is an important crystal form.A thin sheet has for all practical purposes no thickness - that is, the

**geometrical**pattern marked on it will develop the object required after it is bent.Nearly all patterns are the developments of the envelopes of

**geometrical**solids of regular or irregular outlines, few of plane faces; when they are made up of combinations of plane faces, or of faces curved in one plane only, there is no difference in dealing with thin sheets or thick plates.Only common

**geometrical**problems are involved in the case of sheets of sensible thickness, and allowances are made for thickness.Klingenstierna showed from purely

**geometrical**considerations, fully appreciated by Dollond, that the results of Newton's experiments could not be brought into harmony with other universally accepted facts.The graves at Hallstatt were partly inhumation partly cremation; they contained swords, daggers, spears, javelins, axes, helmets, bosses and plates of shields and hauberks, brooches, various forms of jewelry, amber and glass beads, many of the objects being decorated with animals and

**geometrical**designs.From the purely

**geometrical**point of view, a quaternion may be regarded as the quotient of two directed lines in space - or, what comes to the same thing, as the factor, or operator, which changes one directed line into another.Hamilton was led to his great invention by keeping

**geometrical**applications constantly before him while he endeavoured to give a real significance to .,I - i.We will therefore confine ourselves, so far as his predecessors are concerned, to attempts at interpretation which had

**geometrical**applications in view.One

**geometrical**interpretation of the negative sign of algebra was early seen to be mere reversal of direction along a line.Hamilton had

**geometrical**application as his main object; when he realized the quaternion system, he felt that his object was gained, and thenceforth confined himself to the development of his method.The fundamental

**geometrical**conceptions are the point, line and plane.The formula of Dalton would make the pressure increase in

**geometrical**progression for equal increments of temperature.This line is called the boundary of the

**geometrical**shadow, and its construction is based on the assumption that light travels in straight lines (in homogeneous media) and suffers no deviation on meeting an obstacle.An instance of the

**geometrical**shadow is seen when a very small gas-jet is burning in a ground-glass shade near a wall.When there are more luminous points than one, we have only to draw separately the

**geometrical**shadows due to each of the sources, and then superpose them.There will be, in general, portions of all the separate

**geometrical**shadows which overlap one another in some particular regions of the screen.If we were to plug the hole, the whole screen would be in

**geometrical**shadow.The smaller the hole (so far at least as

**geometrical**optics is concerned) the less confused will the picture be.The mere

**geometrical**description and analysis of various types of motion, apart from the consideration of the forces concerned, belongs tt kinematics.As a simple example of the

**geometrical**method of treating statical problems we may consider the equilibrium of a particle on a rough inclined plane.A

**geometrical**proof of this theorem, which is not restricted to a two-dimensional system, is given later (If).Conversely, any six

**geometrical**relations restrict the body in general to one or other of a series of definite positions, none of which can be departed from without violating the conditions in question.In particular, we infer that couples of the same moment in parallel planes are equivalent; and that couples in any two planes may be compounded by

**geometrical**addition of the corresponding vectors.Theory of Mass-S ystems.This is a purely

**geometrical**subject.Again, since the point of the sphere which is in contact with the plane is instantaneously at rest, we have the

**geometrical**relations u+qa=0, v+pa=o, W0, (20) by (12).We have also the

**geometrical**relations x = (a/c) (qz ry), 5 = (a/c) (rx p1), = (a/c) (pyqx).The question was first discussed by Euler (1750); the

**geometrical**representation to be given is due to Poinsot (1851).The case of three degrees of freedom is instructive on account of the

**geometrical**analogies.The radius of the pitch-circle of a wheel is called the

**geometrical**radius; a circle touching the ends of the teeth is called the addendum circle, and its radius the real radius; the difference between these radii, being the projection of the teeth beyond the pitch-surface, is called the addendum.Cudworth's ideas, like Plato's, have "a constant and never-failing entity of their own," such as we see in

**geometrical**figures; but, unlike Plato's, they exist in the mind of God, whence they are communicated to finite understandings.In the first, the implements are rather of copper than of bronze, tin being absent or in small quantities (2 to 3%); the types are common to Syria and Asia Minor as far as the Hellespont, and resemble also the earliest forms in the Aegean and in central Europe; the pottery is all hand-made, with a red burnished surface, gourd-like and often fantastic forms, and simple

**geometrical**patterns incised; zoomorphic art is very rare, and imported objects are unknown.In the second stage, implements of true bronze (9 to io% tin) become common; painted pottery of buff clay with dull black

**geometrical**patterns appears alongside the red-ware; and foreign imports occur, such as Egyptian blue-glazed beads (XIIth-XIIIth Dynasty, 2500-2000 B.C.),1 and cylindrical Asiatic seals (one of Sargon I., 2000 B.C.).2 In the third stage, Aegean colonists introduced the Mycenaean (late Minoan) culture and industries; with new types of weapons, wheel-made pottery, and a naturalistic art which rapidly becomes conventional; gold and ivory are abundant, and glass and enamels are known.Representative art languishes, except a few childish terra-cottas; decorative art becomes once more purely

**geometrical**, but shows only slight affinity with the contemporary**geometrical**art of the Aegean.Gem-engraving and jewelry follow similar lines; pottery-painting for the most part remains

**geometrical**throughout, with crude survivals of Mycenaean curvilinear forms. Those Aegean influences, however, which had been predominant in the later Bronze Age, and had never wholly ceased, revived, as Hellenism matured and spread, and slowly repelled the mixed Phoenician orientalism.The short appendix, in which the attempt is made to present the chief points of the argument in

**geometrical**form, is a forerunner of the Ethics, and was probably written somewhat later than the rest of the book.It is a remnant of the old

**geometrical**style of gardening.These propositions may be derived from the formulae given above, or proved directly by purely

**geometrical**methods.In his theory of graphs, or

**geometrical**representations of algebraic functions, there are valuable suggestions which have been worked out by others.The length of any arc may be determined by

**geometrical**considerations or by the methods of the integral calculus.The

**geometrical**method is also applicable when it is required to determine the caustic c =a C = co c=%a FIG.It is usually the case that the secondary caustic is easier to determine than the caustic, and hence, when determined, it affords a ready means for deducing, the primary caustic. It may be shown by

**geometrical**considerations that the secondary caustic is a curve similar to the first positive pedal of the reflecting curve, of twice the linear dimensions, with respect to the luminous point.The work falls into two parts, which treat of the asymptotes and singularities of algebraical curves respectively; and extensive use is made of the method of counting constants which plays so large a part in modern

**geometrical**researches.From this time Plucker's

**geometrical**researches practically ceased, only to be resumed towards the end of his life.But in the great Sala dell' Asse (or della Torre) abundant traces of Leonardo's own hand were found, in the shape of a decoration of intricate

**geometrical**knot or plait work .combined with natural leafage; the abstract puzzle-pattern, of a kind in which Leonardo took peculiar pleasure, intermingling in cunning play and contrast with a pattern of living boughs and leaves exquisitely drawn in free and vital growth.It is merely a

**geometrical**determination of the conditions necessarily consequent in England, Scotland and Wales, upon a given mean rainfall over many years, upon evaporation and absorption in particular years (both of which he must judge or determine for himself), and upon certain limiting variations of the rainfall, already stated to be the result of numerous records maintained in Great Britain for more than 50 years.An accurate

**geometrical**solution by C. Culmann gives A ?The market hall (1880) surmounted by a clock-tower is in

**geometrical**Decorated style.In a small commonplace book, bearing on the seventh page the date of January 1663/1664, there are several articles on angular sections, and the squaring of curves and " crooked lines that may be squared," several calculations about musical notes,

**geometrical**propositions from Francis Vieta and Frans van Schooten, annotations out of Wallis's Arithmetic of Infinities, together with observations on refraction, on the grinding of " spherical optic glasses," on the errors of lenses and the method of rectifying them, and on the extraction of all kinds of roots, particularly those " in affected powers."Even in commercial transactions, in dealing with sums of money, the statement of an amount often has reference to the last item added rather than to a total; and

**geometrical**measurements are practically ordinal (§ 26).It has been suggested that as many as six objects can be seen at once; but this is probably only the case with few people, and with them only when the objects have a certain

**geometrical**arrangement.When now we pass to

**geometrical**measurement, each " one " is a thing which is itself divisible, and it cannot be said that at any moment we are counting it; it is only when one is completed that we can count it.Thus, while arithmetical numbering refers to units,

**geometrical**numbering does not refer to units but to the intervals between units.Also most fractions cannot be expressed exactly as decimals; and this is also the case for surds and logarithms, as well as for the numbers expressing certain ratios which arise out of

**geometrical**relations.The transition is similar to that which arises in the case of

**geometrical**measurement (§ 26), and it is an essential feature of all reasoning with regard to continuous quantity, such as we have to deal with in real life.Obviously the number of such

**geometrical**or kinematical definitions is infinite.And, assuming the above theory of

**geometrical**imaginaries, a curve such that m of its points are situate in an arbitrary line is said to be of the order m; a curve such that n of its tangents pass through an arbitrary point is said to be of the class n; as already appearing, this notion of the order and class of a curve is, however, due to Gergonne.If, however, the

**geometrical**property requires two or more relations between the coefficients, say A = o, B = o,&c., then we must have between the new coefficients the like relations, A' = o, B' = o, &c., and the two systems of equations must each of them imply the other; when this is so, the system of equations, A = o, B = o, &c., is said to be invariantive, but it does not follow that A, B, &c., are of necessity invariants of u.Coming next to Chasles, the principle of correspondence is established and used by him in a series of memoirs relating to the conics which satisfy given conditions, and to other

**geometrical**questions, contained in the Comptes rendus, t.The principle of correspondence, or say rather the theorem of united points, is a most powerful instrument of investigation, which may be used in place of analysis for the determination of the number of solutions of almost every

**geometrical**problem.The principle in its original form as applying to a right line was used throughout by Chasles in the investigations on the number of the conics which satisfy given conditions, and on the number of solutions of very many other

**geometrical**problems.We have thus a means of

**geometrical**representation for the portions, as well imaginary as real, of any real or imaginary curve.Analogous to the order and class of a plane curve we have the order, rank and class of the system (assumed to be a

**geometrical**one), viz.The 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.Theoretical Astronomy,which may be considered as an extension of

**geometrical**astronomy and includes the determination of the positions and motions of the heavenly bodies by combining mathematical theory with observation.The

**geometrical**concepts just defined are shown in fig.The changes which the aspect of the heaven undergoes, as we travel North and South, are so well known that they need not be described in detail here; but a general statement of them will give a luminous idea of the

**geometrical**co-ordinates we have described.In the Principia Newton made several investigations to determine the effects of these actions; but the

**geometrical**method which he employed could lead only to rude approximations.His supreme merit, however, consisted in the establishment of astronomy on a sound

**geometrical**basis.The Ptolemaic system was, in a

**geometrical**sense, defensible; it harmonized fairly well with appearances, and physical reasonings had not then been extended to the heavens.The outcome of his discoveries was, not only to perfect the

**geometrical**plan of the solar system, but to enhance very materially the predicting power of astronomy.But the direct or

**geometrical**mode of attack has still the preference over any of the indirect plans.A luminous idea of the

**geometrical**relations of the moon, earth and sun will be gained from the figure, by imagining the sun to be moved towards the left, and placed at a distance of 20 ft.We may conclude the ancient history of the lunar theory by saying that the only real progress from Hipparchus to Newton consisted in the more exact determination of the mean motions of the moon, its perigee and its line of nodes, and in the discovery of three inequalities, the representation of which required

**geometrical**constructions increasing in complexity with every step.But the great founder of celestial mechanics employed a

**geometrical**method, ill-adapted to lead to the desired result; and hence his efforts to construct a lunar theory are of more interest as illustrations of his wonderful power and correctness in mathematical reasoning than as germs of new methods of research.Metrical relations between the axes, eccentricity, distance between the foci, and between these quantities and the co-ordinates of points on the curve (referred to the axes and the centre), and focal distances are readily obtained by the methods of

**geometrical**conics or analytically.In fact, whatever theory of light be adopted, there are two vectors to be considered, that are at right angles to one another and connected by purely

**geometrical**relations.By these momentous inductions the

**geometrical**theory of the solar system was perfected, and a hitherto unimagined symmetry was perceived to regulate the mutual relations of its members.The extraordinary advances made by him in this branch of knowledge were owing to his happy method of applying mathematical analysis to physical problems. As a pure mathematician he was, it is true, surpassed in profundity by more than one among his pupils and contemporaries; and in the wider imaginative grasp of abstract

**geometrical**principles he cannot be compared with Fermat, Descartes or Pascal, to say nothing of Newton or Leibnitz.He studied the properties of the cycloid, and attempted the problem of its quadrature; and in the "infinitesimals," which he was one of the first to introduce into

**geometrical**demonstrations, was contained the fruitful germ of the differential calculus.In geometry, and in

**geometrical**crystallography, the term denotes a line which serves to aid the orientation of a figure.During 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.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.Archimedes contributed to the knowledge of these curves by determining the area of the parabola, giving both a

**geometrical**and a mechanical solution, and also by evaluating the ratio of elliptic to circular spaces.This idea that material representation involves a profanation of divine personages, while disallowing all religious art which goes beyond scroll-work, spirals, flourishes and

**geometrical**designs, yet admits to the full of secular art; and accordingly the iconoclastic emperors replaced the holy pictures in churches with frescoes of hunting scenes, and covered their palaces with garden scenes where men were plucking fruit and birds singing amid the foliage.The art of these countries is mainly

**geometrical**, and allows only of monograms crowned with laurels, of peacocks, of animals gambolling amid foliage, of fruit and flowers, of crosses which are either svastikas of Hindu and Mycenaean type, or so lost in enveloping arabesques as to be merely decorative.In mechanics, he made many researches, substituting the notion of the continuity of

**geometrical**displacements for the principle of the continuity of matter.He introduces what are now called the

**geometrical**forms (the row, flat pencil, &c.), and establishes between their elements a one-one correspondence, or, as he calls it, makes them projective.This usually requires large algebraic computations due to the

**geometrical**quantities entering the field equations and equations of motion.Rather than resorting to allegory he defended the literal meaning by arguing that Moses meant

**geometrical**cubits - equal to 6 ordinary cubits.But more importantly, in contrast to numbers and

**geometrical**shapes or intelligible laws, which are also immaterial, life is a living.Not only is the product a stereospecific optical isomer, the reactant is it self a stereospecific

**geometrical**isomer!In AS and A2 Chemistry, we only need to know about

**geometrical**isomerism caused by a C=C bond in the molecule.**Geometrical**optics are a useful tool for calculating reflections from the polygons in a 3D database.The ' hierarchy of optics ' from quantum optics down to paraxial

**geometrical**optics - how does it all fit together?Over all

**geometrical**theorems they would be in complete agreement, only interpreting the words in terms of their respective intuitions.The upper windows are large and have

**geometrical**tracery; so have the tower windows; the lower windows are simpler.The second (a

**geometrical**proof) is expounded in Props.The stranger, Isaac Beeckman, principal of the college of Dort, offered to do so into Latin, if the inquirer would bring him a solution of the problem, - for the advertisement was one of those challenges which the mathematicians of the age were accustomed to throw down to all corners, daring them to discover a

**geometrical**mystery known as they fancied to themselves alone.These doctrines of inertia, and of the composite character of curvilinear motion, were scarcely apprehended even by Kepler or Galileo; but they follow naturally from the

**geometrical**analysis of Descartes.The nature of logarithms is explained by reference to the motion of points in a straight line, and the principle upon which they are based is that of the correspondence of a

**geometrical**and an arithmetical series of numbers.The general style of works of the Cosmati school is Gothic in its main lines, especially in the elaborate altar-canopies, with their pierced

**geometrical**tracery.In 1868 he was awarded the Steiner prize of the Berlin Academy for a

**geometrical**memoir, Sur quelques problemes cubiques et biquadratiques.Maclaurin's object was to found the doctrine of fluxions on

**geometrical**demonstration, and thus to answer all objections to its method as being founded on false reasoning and full of mystery.If, however, two compounds only differ with regard to the spatial arrangement of the atoms, the physical properties may be (I) for the most part identical, differences, however, being apparent with regard to the action of the molecules on polarized light, as is the case when the configuration is due to the presence of an asymmetric atom (optical isomerism); or (2) both chemical and physical properties may be different when the configuration is determined by the disposition of the atoms or groups attached to a pair of doubly-linked atoms, or to two members of a ring system (

**geometrical**isomerism or allo-isomerism).His

**geometrical**discoveries are of great value, his Die freie Perspective (1759-1774) being a work of great merit.The 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.The difficulty first arises in elementary mensuration, where it is partly met by associating arithmetical and

**geometrical**measurement with the cardinal and the ordinal aspects of number respectively (see Arithmetic).It was at last realized that the laws of algebra do not depend for their validity upon any particular interpretation, whether arithmetical,

**geometrical**or other; the only question is whether these laws do or do not involve any logical contradiction.To assist his lectures on astronomy he constructed elaborate globes of the terrestrial and celestial spheres, on which the course of the planets was marked; for facilitating arithmetical and perhaps

**geometrical**processes he constructed an abacus with twenty-seven divisions and a thousand counters of horn.The question now arises whether the sensation produced by a periodic disturbance can be analysed in correspondence with this

**geometrical**analysis.Thus the number lox in the arithmetical column corresponds to 10 8 (1.0001) x in the

**geometrical**column; the intermediate numbers being obtained by interpolation.If we divide the numbers in the

**geometrical**column by lo g the correspondence is between lox and (I 000l) x, and the table then becomes one of antilogarithms, the base being (1.0001) 1 / 10, viz.He was aware that the wellknown

**geometrical**methods of the ancients would clothe his new creations in a garb which would appear less strange and uncouth to those not familiar with the new method.His invention of the proportional compass or sector - an implement still used in

**geometrical**drawing - dates from 1597; and about the same time he constructed the first thermometer, consisting of a bulb and tube filled with air and water, and terminating in a vessel of water.Kepler'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).His investigations are distinguished by their great generality, by the fertility of his resources, and by such a rigour in his proofs that he has been considered the greatest

**geometrical**genius since the time of Apollonius.This page outlines the

**geometrical**theory underlying the spar gage and then gives guidance on construction.Lots of tessellating

**geometrical**shapes, do you think that there is some significance?I lifted the well cover which was shaped into the sacred

**geometrical**shape known as the vesica piscis.Designed in the

**geometrical**Gothic style, it has paired clerestory windows with tracery.When southwestern design is mentioned most people think of turquoise and Aztec

**geometrical**patterns.The flags are mounted on a bold red background that is easily seen, and the

**geometrical**design is easily replicated in different sizes.Elements that are easily identifiable in furniture pieces are elegant,

**geometrical**designs and angular pieces.Shaved patterns along the sides or back of head, including

**geometrical**shapes, words, or symbols.A Herkimer diamond, however, is formed in the earth with a natural

**geometrical**shape, giving it the appearance of having been cut by a jeweler.This era, which encompassed the 1920s and 1930s, was all about

**geometrical**patterns and elaborate filigree work.For during the year that elapsed before he left Swabia (and whilst he sojourned at Neuburg and Ulm), and amidst his

**geometrical**studies, he would fain have gathered some knowledge of the mystical wisdom attributed to the Rosicrucians; but the Invisibles, as they called themselves, kept their secret.And the algebraists or arithmeticians of the 16th century, such as Luca Pacioli (Lucas de Borgo), Geronimo or Girolamo Cardano (1501-1576), and Niccola Tartaglia (1506-1559), had used

**geometrical**constructions to throw light on the solution of particular equations.Elie de Beaumont, in his speculations on the relation between the direction of mountain ranges and their geological age and character, was feeling towards a comprehensive theory of the forms of crustal relief; but his ideas were too

**geometrical**, and his theory that the earth is a spheroid built up on a rhombic dodecahedron, the pentagonal faces of which determined the direction of mountain ranges, could not be proved.'In his youth he went to the continent and taught mathematics at Paris, where he published or edited, between the years 1612 and 1619, various

**geometrical**and algebraical tracts, which are conspicuous for their ingenuity and elegance.In 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.The most beautiful portion of the mosque, however, still exists in the prayer chamber of Hakim, where are to be found the earliest examples of the cusped arch and the origin of many of the

**geometrical**patterns in stucco at the Alhambra.Among the contents of this book we simply mention a trigonometrical chapter, in which the words sinus versus arcus occur, the approximate extraction of cube roots shown more at large than in the Liber abaci, and a very curious problem, which nobody would search for in a

**geometrical**work, viz.He not only freed it from all trammels of

**geometrical**construction, but by the introduction of the symbol b gave it the efficacy of a new calculus.The main work of Descartes, so far as algebra was concerned, was the establishment of a relation between arithmetical and

**geometrical**measurement.This involved not only the

**geometrical**interpretation of negative quantities, but also the idea of continuity; this latter, which is the basis of modern analysis, leading to two separate but allied developments, viz.While mensuration is concerned with the representation of

**geometrical**magnitudes by numbers, graphics is concerned with the representation of numerical quantities by**geometrical**figures, and particularly by lengths.The development is based on the necessity of being able to represent

**geometrical**magnitude by arithmetical magnitude; and it may be regarded as consisting of three stages.The 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.It is remarkable that Mobius employs the symbols AB, ABC, Abcd In Their Ordinary

**Geometrical**Sense As Lengths, Areas And Volumes, Except That He Distinguishes Their Sign; Thus Ab = Ba, Abc= Acb, And So On.Notwithstanding the prolixity of writers and the number of the writings, all attempts at extracting an algebraic analysis from their

**geometrical**theorems and problems have been fruitless, and it is generally conceded that their analysis was**geometrical**and had little or no affinity to algebra.Investigation of the writings of Indian mathematicians has exhibited a fundamental distinction between the Greek and Indian mind, the former being pre-eminently

**geometrical**and speculative, the latter arithmetical and mainly practical.Although the foundations of the

**geometrical**resolution of cubic equations are to be ascribed to the Greeks (for Eutocius assigns to Menaechmus two methods of solving the equation x 3 = a and x 3 = 2a 3), yet the subsequent development by the Arabs must be regarded as one of their most important achievements.He improved the methods for solving equations, and devised

**geometrical**constructions with the aid of the conic sections.He possessed clear ideas of indices and the generation of powers, of the negative roots of equations and their

**geometrical**interpretation, and was the first to use the term imaginary roots.So 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.The

**geometrical**interpretation of imaginary quantities had a far-reaching influence on the development of symbolic algebras.If the point under consideration be so far away from the

**geometrical**shadow that a large number of the earlier zones are complete, then the illumination, determined sensibly by the first zone, is the same as if there were no obstruction at all.If, on the other hand, the point be well immersed in the

**geometrical**shadow, the earlier zones are altogether missing, and, instead of a series of terms beginning with finite numerical magnitude and gradually diminishing to zero, we have now to deal with one of which the terms diminish to zero at both ends.At the point 0 the intensity is one-quarter of that of the entire wave, and after this point is passed, that is, when we have entered the

**geometrical**shadow, the intensity falls off gradually to zero, without fluctuations.Descartes strengthened these views, both by experiments and

**geometrical**investigations, in his Meteors (Leiden, 1637).Only when we have admitted the conception of the infinitely small, and the resulting

**geometrical**progression with a common ratio of one tenth, and have found the sum of this progression to infinity, do we reach a solution of the problem.This treatise is in two books, dedicated to Dositheus, and deals with the dimensions of spheres, cones, "solid rhombi" and cylinders, all demonstrated in a strictly

**geometrical**method.The whole conception of force may disappear from a theory of the universe; and we can adopt a

**geometrical**definition of motion as the shifting of one body from the neighbourhood of those bodies which immediately touch it, and which are assumed to be at rest, to the neighbourhood of other bodies.From the descriptive or topographical point of view,

**geometrical**form alone should be con- Land sidered; but the origin and geological structure of forms. land forms must in many cases be taken into account when dealing with the function they exercise in the control of mobile distributions.A new

**geometrical**style of decoration like that of contemporary Greece largely supplants the Minoan models.He was thus led to conclude that chemistry is a branch of applied mathematics and to endeavour to trace a law according to which the quantities of different bases required to saturate a given acid formed an arithmetical, and the quantities of acids saturating a given base a

**geometrical**, progression.His first contributions to mathematical physics were two papers published in 1873 in the Transactions of the Connecticut Academy on "Graphical Methods in the Thermodynamics of Fluids," and "Method of

**Geometrical**Representation of the Thermodynamic Properties of Substances by means of Surfaces."This 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.I could not follow with my eyes the

**geometrical**figures drawn on the blackboard, and my only means of getting a clear idea of them was to make them on a cushion with straight and curved wires, which had bent and pointed ends.The

**geometrical**diagrams were particularly vexing because I could not see the relation of the different parts to one another, even on the cushion.