# Cartesian Sentence Examples

Its

**cartesian**equation is x 3 -1-y 3 =3axy.Pascal and other members of Port Royal openly expressed their doubts about the place allowed to God in the system; the adherents of Gassendi met it by resuscitating atoms; and the Aristotelians maintained their substantial forms as of old; the Jesuits argued against the arguments for the being of God, and against the theory of innate ideas; whilst Pierre Daniel Huet (1630-1721), bishop of Avranches, once a

**Cartesian**himself, made a vigorous onslaught on the contempt in which his former comrades held literature and history, and enlarged on the vanity of all human aspirations after rational truth.The

**cartesian**equation referred to the axis and directrix is y=c cosh (x/c) or y = Zc(e x / c +e x / c); other forms are s = c sinh (x/c) and y 2 =c 2 -1-s 2, being the arc measured from the vertex; the intrinsic equation is s = c tan The radius of curvature and normal are each equal to c sec t '.The most important formulae are those which correspond to the use of rectangular

**Cartesian**co-ordinates.In 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.Analytically, the

**Cartesian**equation to a coaxal system can be written in the form x 2 + y 2 + tax k 2 = o, where a varies from member to member, while k is a constant.Essentially, therefore, Descartes's process is that known later as the process of isoperimeters, and often attributed wholly to Schwab.2 In 16J5 appeared the Arithmetica Infinitorum of John Wallis, where numerous problems of quadrature are dealt with, the curves being now represented in

**Cartesian**co-ordinates, and algebra playing an important part.The generality of treatment is indeed remarkable; he gives as the fundamental property of all the conics the equivalent of the

**Cartesian**equation referred to oblique axes (consisting of a diameter and the tangent at its extremity) obtained by cutting an oblique circular cone in any manner, and the axes appear only as a particular case after he has shown that the property of the conic can be expressed in the same form with reference to any new diameter and the tangent at its extremity.Apollonius' genius takes its highest flight in Book v., where he treats of normals as minimum and maximum straight lines drawn from given points to the curve (independently of tangent properties), discusses how many normals can be drawn from particular points, finds their feet by construction, and gives propositions determining the centre of curvature at any point and leading at once to the

**Cartesian**equation of the evolute of any conic.Even the

**Cartesian**school, as it came more and more to feel the difficulty of explaining the interaction of body and mind, and, indeed, any efficient causation whatever, gradually tended to the hypothesis that the real cause is God, who, on the occasion of changes in body, causes corresponding changes in mind, and vice versa.AdvertisementFurther, he explained the old

**Cartesian**difficulty of the relation of body and mind by transforming the Spinozistic parallelism of extension and thought into a parallelism between the motions of bodies and the perceptions of their monads; motions always proceeding from motions, and perceptions from perceptions; bodies acting according to efficient causes, and souls according to final causes by appetition, and as if one influenced the other without actually doing so.In order to prove this novel conclusion he started afresh from the

**Cartesian**" I think " in the Kantian form of the synthetic unity of apperception acting by a priori categories; but instead of allowing, with all previous metaphysicians, that the Ego passively receives sensations from something different, and not contenting himself with Kant's view that the Ego, by synthetically combining the matter of sensations with a priori forms, partially constructs objects, and therefore Nature as we know it, he boldly asserted that the Ego, in its synthetic unity, entirely constructs things; that its act of spontaneity is not mere synthesis of passive sensations, but construction of sensations into an object within itself; and that therefore understanding makes as well as shapes Nature.But his main reliance is on the passage in the Kritik, where Kant, speaking of the

**Cartesian**difficulty of communication between body and soul, suggests that, however body and soul appear to be different in the phenomena of outer and inner sense, what lies as thing in itself at the basis of the phenomena of both may perhaps be not so heterogeneous (ungleichartig) after all.In France, the home of

**Cartesian**realism, after the vicissitudes of sensationalism and materialism, which became connected in French the French mind with the Revolution, the spirit of Descartes revived in the 19th century in the spiritualistic realism of Victor Cousin.Chouet (1642-1731) the

**Cartesian**, and attended the theological lectures of P. Mestrezat, Franz Turretin and Louis Tronchin (1629-1705).AdvertisementDaniel also wrote a by no means successful reply to Pascal's Provincial Letters, entitled Entretiens de Cleanthe provinciales (1694); two treatises on the

**Cartesian**theory as to the intelligence of the lower animals, and other works.The

**cartesian**parabola is a cubic curve which is also known as the trident of Newton on account of its three-pronged form.The system of Rohault was founded entirely upon

**Cartesian**principles, and was previously known only through the medium of a rude Latin version.Though Clarke can thus be defended against this and similar criticism, his work as a whole can be regarded only as an attempt to present the doctrines of the

**Cartesian**school in a form which would not shock the conscience of his time.As to syllogism specifically, Locke in a passage, 8 which has an obviously

**Cartesian**ring, lays down four stages or degrees of reasoning, and points o ut that syllogism serves us in but one of these, and that not the all-important one of finding the intermediate ideas.AdvertisementA fundamental contrast to the school of Bacon and of Locke is afforded by the great systems of reason, owning

**Cartesian**inspira tion, which are identified with the names of Spinoza and Leibnitz.Quaternions (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.Neither of these men professed to employ the calculus itself, but they recognized fully the extraordinary clearness of insight which is gained even by merely translating the unwieldy

**Cartesian**expressions met with in hydrokinetics and in electrodynamics into the pregnant language of quaternions.On leaving the Erasmus school at Rotterdam he gave proof of his ability by an Oratio scholastica de medicina (1685), and at Leiden University in 1689 he maintained a thesis De brutorum operationibus, in which he advocated the

**Cartesian**theory of automatism among animals.The problem of determining the possible configurations of equilibrium of a system of particles subject to extraneous forces which are known functions of the positions of the particles, and to internal forces which are known functions of the distances of the pairs of particles between which they act, is in general determinate For if n be the number of particles, the 3n conditions of equilibrium (three for each particle) are equal in number to the 351

**Cartesian**(or other) co-ordinates of the particles, which are to be found.AdvertisementSince the four co-ordinates (

**Cartesian**or other) of these two points are connected by the relation which expresses the invariability of the length AB, it is plain that virtually three inde pendent elements are re quired and suffice to specify the position of the lamina.We may, for instance, employ the three

**Cartesian**co-ordinates of a particular point 0 of the body, and three angular co-ordinates which express the orientation of the body with respect to 0.Proceeding to the general motion of a rigid body in two dimensions we may take as the three co-ordinates of the body the rectangular

**Cartesian**co-ordinates x, y of the mass-centre G and the angle C through which the body has turned from some standard position.It is implied in the above description of the system that the

**Cartesian**co-ordinates x, y, z of any particle of the system are known functions of the qs, varying in form (of course) from particle to particle.This solution, taken by itself, represents a motion in which each particle of the system (since its displacements parallel to

**Cartesian**co-ordinate axes are linear functions of the qs) executes a simple vibration of period 21r/u.Originally meeting in all probability for more thoroughgoing study of the

**Cartesian**philosophy, they looked naturally to Spinoza for guidance, and by and by we find him communicating systematic drafts of his own views to the little band of friends and students.The term "Nature" is put more into the foreground in the Treatise, a point which might be urged as evidence of Bruno's influence - the dialogues, moreover, being specially concerned to establish the unity, infinity and selfcontainedness of Nature 2; but the two opposed

**Cartesian**attributes, thought and extension, and the absolutely infinite substance whose attributes they are - substance constituted by infinite attributes - appear here as in the Ethics.Although the systematic framework of the thought and the terminology used are both derived from the

**Cartesian**philosophy, the intellectual milieu of the time, the early work enables us, better than the Ethics to realize that the inspiration and starting-point of his thinking is to be found in the religious speculations of his Jewish predecessors.The histories of philosophy may quite correctly describe his theory as the logical development of Descartes's doctrines of the one Infinite and the two finite substances, but Spinoza himself was never a

**Cartesian**.He was the friend of regular correspondent - a third of the letters preserved to us are to or from him; and it appears from his first letter that their talk on this occasion was "on God, on infinite extension and thought, on the difference and the agreement of these attributes, on the nature of the union of the human soul with the body, as well as concerning the principles of the

**Cartesian**and Baconian philosophies."At the request of his friends he devoted a fortnight to applying the same method to the first or metaphysical part of Descartes's philosophy, and the sketch was published in 1663, with an appendix entitled Cogitata metaphysica, still written from a

**Cartesian**standpoint (defending, for example, the freedom of the will), but containing hints of his own doctrine.If the ratio of the radii be as I to 4, we obtain the four-cusped hypocycloid, which has the simple

**cartesian**equation x 2'3+ y 213 = a 21 '.The

**cartesian**equation was first given by Wilhelm Gottfried Leibnitz (Ada eruditorum, 1686) in the form y = (2xx 2)-1.The

**cartesian**equation in terms similar to those used above is x = a6+b sin 0; y=a-b cos 0, where a is the radius of the generating circle and b the distance of the carried point from the centre of the circle.The

**cartesian**equation, referred to the fixed diameter and the tangent at B as axes may be expressed in the forms x= a6, y=a(I -cos 0) and y-a=a sin (x/afir); the latter form shows that the locus is the harmonic curve.The

**Cartesian**equation to the caustic produced by reflection at a circle of rays diverging from any point was obtained by Joseph Louis Lagrange; it may be expressed in theform 1(4,2_ a2) (x 2+ y2) - 2a 2 cx - a 2 c 2 1 3 = 2 7 a4c2y2 (x2 + y2 - c2)2, where a is the radius of the reflecting circle, and c the distance of the luminous point from the centre of the circle.When the refracting curve is a circle and the rays emanate from any point, the locus of the secondary caustic is a

**Cartesian**oval, and the evolute of this curve is the required diacaustic. These curves appear to have been first discussed by Gergonne.The

**cartesian**equation, if A be taken as origin and AB (= 2a) for the axis of x, is xy 2 =4a2(2a - x).The Greek geometers were perfectly familiar with the property of an ellipse which in the

**Cartesian**notation is x 2 /a 2 +y 2 /b 2 =1, the equation of the curve; but it was as one of a number of properties, and in no wise selected out of the others for the characteristic property of the curve.It is convenient to use these rather than

**Cartesian**co-ordinates.The circular cubic and the bicircular quartic, together with the

**Cartesian**(being in one point of view a particular case thereof), are interesting curves which have been much studied, generally, and in reference to their focal properties.The analytical theory by

**Cartesian**co-ordinates was first considered by Alexis Claude Clairaut, Recherches sur les courbes et double courbure (Paris, 1731).The

**cartesian**equation, when A is the origin and AB = 2a, is y 2 (2a - x) =x 8; the polar equation is r= 2a sin 0 tan 0.This reasoning, in which Anselm partially anticipated the

**Cartesian**philosophers, has rarely seemed satisfactory.He did not succeed either in stemming the tide of expense, nor in his administration, being in no way in advance of his age, and not perceiving that decisive reform could not be achieved by a government dealing with the nation as though it were inert and passive material, made to obey and to payS Like a good

**Cartesian**he conceived of the state as an immense machine, every portion of which should receive its impulse from outsidethat is from him, Colbert.John Wallis, in addition to translating the Conics of Apollonius, published in 1655 an original work entitled De sectionibus conicis nova methodo expositis, in which he treated the curves by the

**Cartesian**method, and derived their properties from the definition in piano, completely ignoring the connexion between the conic sections and a cone.A method of generating conics essentially the same as our modern method of homographic pencils was discussed by Jan de Witt in his Elementa linearum curvarum (1650); but he treated the curves by the

**Cartesian**method, and not synthetically.With the

**Cartesian**movement as a whole he shows little acquaintance and no sympathy, and his own philosophic conception is never brought into relation with the systematic treatment of metaphysical problems characteristic of the**Cartesian**method.The transition from the

**Cartesian**movement to this second stage of modern thought had doubtless been natural and indeed necessary.Calculates the relaxing density contribution to the derivative of the energy wrt

**Cartesian**coordinates.The argument is that the

**Cartesian**approach, to start with self evident premises and to derive rational conclusions from them is too limited.Such a scheme builds conscious subjectivity into the universe in a more integral way than

**Cartesian**dualism seems to do.In the Grammar, Newman makes his case for a radically new understanding of human reason, rejecting both

**Cartesian**rationalism and Lockean empiricism.The anonymous objections are very much the statement of common-sense against philosophy; those of Caterus criticize the

**Cartesian**argument from the traditional theology of the church; those of Arnauld are an appreciative inquiry into the bearings and consequences of the meditations for religion and morality; while those of Hobbes (q.v.) and Gassendi - both somewhat senior to Descartes and with a dogmatic system of their own already formed - are a keen assault upon the spiritualism of the**Cartesian**position from a generally " sensational " standpoint.Such are the half-hearted attempts at consistency in

**Cartesian**thought, which eventually culminate in the pantheism of Spinoza (see Cartesianism).Yet in 1675 the university of Angers was empowered to repress all

**Cartesian**teaching within its domain, and actually appointed a commission charged to look for such heresies in the theses and the students' note-books of the college of Anjou belonging to the Oratory.But this doctrine was a criticism and a divergence, no less than a consequence, from the principles in Descartes; and it brought upon Malebranche the opposition, not merely of the

**Cartesian**physicists, but also of Arnauld, Fenelon and Bossuet, who found, or hoped to find, in the Meditations, as properly understood, an ally for theology.Geulincx carried out to their extreme consequences the irreconcilable elements in the

**Cartesian**metaphysics, and his works have the peculiar value attaching to the vigorous development of a one-sided principle.The

**cartesian**equation to a parabola which touches the coordinate axes is 1 / ax+'1 / by= i, and the polar equation when the focus is the pole and the axis the initial line is r cos 2 6/2 = a.It is to her that the Principles of Philosophy were dedicated; and in her alone, according to Descartes, were united those generally separated talents for metaphysics and for mathematics which are so characteristically co-operative in the

**Cartesian**system.The contempt of aesthetics and erudition is characteristic of the most typical members of what is known as the

**Cartesian**school, especially Malebranche.The first public teacher of

**Cartesian**views was Henri Renery, a Belgian, who at Deventer and afterwards at Utrecht had introduced the new philosophy which he had learned Spread of from personal intercourse with Descartes.In 1639 he published a series of arguments against atheism, in which the

**Cartesian**views were not obscurely indicated as perilous for the faith, though no name was mentioned.There the

**Cartesian**innovations had found a patron in Adrian Heerebord, and were openly discussed in theses and lectures.To attach a clear and definite meaning to the

**Cartesian**doctrine of God, to show how much of it comes from the Christian theology and how much from the logic of idealism, how far the conception of a personal being as creator and preserver mingles with the pantheistic conception of an infinite and perfect something which is all in all, would be to go beyond Descartes and to ask for a solution of difficulties of which he was 1 Ouvres, vi.In both these doctrines of a priori science Descartes has not been subverted, but, if anything, corroborated by the results of experimental physics; for the so-called atoms of chemical theory already presuppose, from the

**Cartesian**point of view, certain aggregations of the primitive particles of matter.But the

**Cartesian**theory, like the later speculations of Kant and Laplace, proposes to give a hypothetical explanation of the circumstances and motions which in the normal course of things led to the state of things required by the law of attraction.It should be added that the modern theory of vortex-atoms (Lord Kelvin's) to explain the constitution of matter has but slight analogy with

**Cartesian**doctrine, and finds a parellel, if anywhere, in a modification of that doctrine by Malebranche.The chief names in this advanced theology connected with

**Cartesian**doctrines are Ludwig Meyer, the friend and editor of Spinoza, author of a work termed Philosophia scripturae interpres (1666); Balthasar Bekker, whose World Bewitched helped to discredit the superstitious fancies about the devil; and Spinoza, whose Tractatus theologico-politicus is in some respects the classical type of rational criticism up to the present day.The Chateau of the duc de Luynes, the translator of the Meditations, was the home of a

**Cartesian**club, that discussed the questions of automatism and of the composition of the sun from filings and parings, and rivalled Port Royal in its vivisections.On his visit to Toulouse in 1665, with a mission from the

**Cartesian**chiefs, his lectures excited boundless interest; ladies threw themselves with zeal and ability into the study of philosophy; and Regis himself .was made the guest of the civic corporation.From the real or fancied rapprochements between Cartesianism and Jansenism, it became for a while impolitic, if not dangerous, to avow too loudly a preference for

**Cartesian**theories.And before 1725, readings, both public and private, were given from

**Cartesian**texts in some of the Parisian colleges.Fortunately the

**Cartesian**method had already done its service, even where the theories were rejected.In Germany a few

**Cartesian**lecturers taught at Leipzig and Halle, but the system took no root, any more than in Switzerland, where it had a brief reign at Geneva after 1669.At Naples there grew up a

**Cartesian**school, of which the best known members are Michel Angelo Fardella (1650-1708) and Cardinal Gerdil (1718-1802), both of whom, however, attached themselves to the characteristic views of Malebranche.He is the author of several works, amongst others a system of

**Cartesian**philosophy, where a chapter on " Angels " revives the methods of the schoolmen.His chief opponent was Samuel Parker (1640-1688), bishop of Oxford, who, in his attack on the irreligious novelties of the

**Cartesian**, treats Descartes as a fellow-criminal in infidelity with Hobbes and Gassendi.Rohault's version of the

**Cartesian**physics was translated into English; and Malebranche found an ardent follower in John Norris (1667-171 I).Forbes communicated to the Royal Society of Edinburgh a short paper of his on a mechanical method of tracing

**Cartesian**ovals.Embracing the whole philosophic movement under the name of "the

**Cartesian**system," Reid detects its fundamental error in the unproved assumption shared by these thinkers "that all the objects of my knowledge are ideas in my own mind."After travelling in France and England, he studied the

**Cartesian**philosophy under John Racy at Leiden.Its

**cartesian**equation, when the line joining the two fixed points is the axis of x and the middle point of this line is the origin, is (x 2 + y 2)2 = 2a 2 (x 2 - y 2) and the polar equation is r 2 = 2a 2 cos 20.Also, as the

**Cartesian**geometry shows, all the relations between points are expressible in terms of geometric quantities.Glanvill's first work (a passage in which suggested the theme of Matthew Arnold's Scholar Gipsy), The Vanity of Dogmatizing, or Confidence in Opinions, manifested in a Discourse of the shortness and uncertainty of our Knowledge, and its Causes, with Reflexions on Peripateticism, and an Apology for Philosophy (1661), is interesting as showing one special direction in which the new method of the

**Cartesian**philosophy might be developed.But this example, combined with the

**Cartesian**principles, set many active and ingenious spirits to work to reconstruct the whole of medicine on a physiological or even a mechanical basis - to endeavour to form what we should now call physiological or scientific medicine.At a time when the

**Cartesian**system of vortices universally prevailed, he found it necessary to investigate that hypothesis, and in the course of his investigations he showed that the velocity of any stratum of the vortex is an arithmetical mean between the velocities of the strata which enclose it; and from this it evidently follows that the velocity of a filament of water moving in a pipe is an arithmetical mean between the velocities of the filaments which surround it.The

**cartesian**equation is y=a cos /2a.This involves the use of

**Cartesian**co-ordinates, and leads to important general formulae, such as Simpson's formula.