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cycloid

The two treatises on the cycloid and on the cissoid, &c., and the Mechanica contain many results which were then new and valuable.

46Accordingly, by the spring of 1660, he had managed to put his criticism and assertions into five dialogues under the title Examinatio et emendatio mathematicae hodiernae qualis explicatur in libris Johannis Wallisii, with a sixth dialogue so called, consisting almost entirely of seventy or more propositions on the circle and cycloid.'

11There has been some discussion as to the fairness of the treatment accorded by Pascal to his rivals, but no question of the fact that his initiative led to a great extension of our knowledge of the properties of the cycloid, and indirectly hastened the progress of the differential calculus.

11The cycloid is the simplest member of the class of curves known as roulettes.

11The cycloid was a famous curve in those days; it had been discussed by Galileo, Descartes, Fermat, Roberval and Torricelli, who had in turn exhausted their skill upon it.

00Pascal solved the hitherto refractory problem of the general quadrature of the cycloid, and proposed and solved a variety of others relating to the centre of gravity of the curve and its segments, and to the volume and centre of gravity of solids of revolution generated in various ways by means of it.

00It may be noticed that if the scales of x and be properly adjusted, the curve of positions in the present problem is the portion of a cycloid extending from a vertex to a cusp.

00In the case of a particle oscillating under gravity on a smooth cycloid from rest at the cusp the hotlograph is a circle through the pole, described with constant velocity.

00Similarly, the corresponding epitrochoids will exhibit three loops or nodes (curve b), or assume the form shown in the curve; c. It is interesting to compare the forms of these curves with the three forms of the cycloid.

00CYCLOID (from Gr.

00The name cycloid is now restricted to the curve described when the tracing-point is on the circumference of the circle; if the point is either within or without the circle the curves are generally termed trochoids, but they are also known as the prolate and curtate cycloids respectively.

00No mention of the cycloid has been found in writings prior to the 15th century.

00Evangelista Torricelli, in the first regular dissertation on the cycloid (De dimensione cycloidis, an appendix to his De dimensione parabolae, 1644), states that his friend and tutor Galileo discovered the curve about 1599.

00He there shows that the cycloid was investigated by Carolus Bovillus about r 500, and by Cardinal Cusanus (Nicolaus de Cusa) as early as 1451.

00Many other mathematicians have written on the cycloid - Blaise Pascal, W.

00A famous period in the history of the cycloid is marked by a bitter controversy which sprang up between Descartes and Roberval.

00Among other early writers on the cycloid were Phillippe de Lahire (1640-1718) and Francois Nicole (1683-1758).

00The mechanical properties of the cycloid were investigated by Christiaan Huygens, who proved the curve to be tautochronous.

00His enquiries into evolutes enabled him to prove that the evolute of a cycloid was an equal cycloid, and by utilizing this property he constructed the isochronal pendulum generally known as the cycloidal pendulum.

00The method by which the cycloid is generated shows that it consists of an infinite number of cusps placed along the fixed line and separated by a constant distance equal to the circumference of the rolling circle.

00The b co-ordinates of any point R on the a ?/' t11®V1 a cycloid are expressible in the form x=a(8-}-sin 0); y=a (I -cos 0), M where the co-ordinate axes are the tangent at the vertex 0 and the axis of the curve, a is the radius of the generating circle, and 0 the angle R'CO, where RR' is parallel to LM and C is the centre of the circle in its symmetric position.

00The intrinsic P equation is s =4a sin 4,, and the equation to the evolute is s= 4a cos 1P, which proves the evolute to be a similar cycloid placed as in fig.

00the original cycloid.

00van Schooten, and more completely by John Wallis, who showed that they possessed properties similar to those of the common cycloid.

001, curve c. The companion to the cycloid is a curve so named on account of its similarity of construction, form and equation to the common cycloid.

00Draw any line DE perpendicular to AB and meeting the circle in E, and take a point P on DE such that the line DP =arc BE; then the locus of P is the companion to the cycloid.

00He announced that the curve required in the first problem must be a cycloid, and he gave a method of determining it.

00Among these subjects were the transit of Mercury, the Aurora Borealis, the figure of the earth, the observation of the fixed stars, the inequalities in terrestrial gravitation, the application of mathematics to the theory of the telescope, the limits of certainty in astronomical observations, the solid of greatest attraction, the cycloid, the logistic curve, the theory of comets, the tides, the law of continuity, the double refraction micrometer, various problems of spherical trigonometry, &c. In 1742 he was consulted, with other men of science, by the pope, Benedict XIV., as to the best means of securing the stability of the dome of St Peter's, Rome, in which a crack had been discovered.

00The publication amongst Torricelli's Opera geometrica (Florence, 1644) of a tract on the properties of the cycloid involved him in a controversy with G.

00He 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.

00cycloid scales but have comb-like teeth on their overlapping edge.

00When the directive force is constant, the curve is a cycloid; under other conditions, spirals and other curves are described (see Mechanics) .

00The determination of the true relation between the length of a pendulum and the time of its oscillation; the invention of the theory of evolutes; the discovery, hence ensuing, that the cycloid is its own evolute, and is strictly isochronous; the ingenious although practically inoperative idea of correcting the "circular error" of the pendulum by applying cycloidal cheeks to clocks - were all contained in this remarkable treatise.

00- Burrowing snakes, mostly small, which have the body covered with smooth, shiny, uniform cycloid scales The teeth are restricted to the small maxillary bones.

00The two treatises on the cycloid and on the cissoid, &c., and the Mechanica contain many results which were then new and valuable.

00The vermiform body is covered with cycloid imbricating scales, devoid of osteoderms. Limbs and even their arches are absent, excepting a pair of flaps which represent the hind-limbs in the males.

00Many fish inhabited the Carboniferous seas and most of these were Elasmobranchs, sharks with crushing pavement teeth (Psammodus), adapted for grinding the shells of brachiopods, crustaceans, &c. Other sharks had piercing teeth (Cladoselache and Cladodus); some, the petalodonts, had peculiar cycloid cutting teeth.

00Accordingly, by the spring of 1660, he had managed to put his criticism and assertions into five dialogues under the title Examinatio et emendatio mathematicae hodiernae qualis explicatur in libris Johannis Wallisii, with a sixth dialogue so called, consisting almost entirely of seventy or more propositions on the circle and cycloid.'

00The cycloid was a famous curve in those days; it had been discussed by Galileo, Descartes, Fermat, Roberval and Torricelli, who had in turn exhausted their skill upon it.

00Pascal solved the hitherto refractory problem of the general quadrature of the cycloid, and proposed and solved a variety of others relating to the centre of gravity of the curve and its segments, and to the volume and centre of gravity of solids of revolution generated in various ways by means of it.

00There has been some discussion as to the fairness of the treatment accorded by Pascal to his rivals, but no question of the fact that his initiative led to a great extension of our knowledge of the properties of the cycloid, and indirectly hastened the progress of the differential calculus.

00It may be noticed that if the scales of x and be properly adjusted, the curve of positions in the present problem is the portion of a cycloid extending from a vertex to a cusp.

00In the case of a particle oscillating under gravity on a smooth cycloid from rest at the cusp the hotlograph is a circle through the pole, described with constant velocity.

00Similarly, the corresponding epitrochoids will exhibit three loops or nodes (curve b), or assume the form shown in the curve; c. It is interesting to compare the forms of these curves with the three forms of the cycloid.

00CYCLOID (from Gr.

00The name cycloid is now restricted to the curve described when the tracing-point is on the circumference of the circle; if the point is either within or without the circle the curves are generally termed trochoids, but they are also known as the prolate and curtate cycloids respectively.

00The cycloid is the simplest member of the class of curves known as roulettes.

00No mention of the cycloid has been found in writings prior to the 15th century.

00Evangelista Torricelli, in the first regular dissertation on the cycloid (De dimensione cycloidis, an appendix to his De dimensione parabolae, 1644), states that his friend and tutor Galileo discovered the curve about 1599.

00He there shows that the cycloid was investigated by Carolus Bovillus about r 500, and by Cardinal Cusanus (Nicolaus de Cusa) as early as 1451.

00Many other mathematicians have written on the cycloid - Blaise Pascal, W.

00A famous period in the history of the cycloid is marked by a bitter controversy which sprang up between Descartes and Roberval.

00Among other early writers on the cycloid were Phillippe de Lahire (1640-1718) and Francois Nicole (1683-1758).

00The mechanical properties of the cycloid were investigated by Christiaan Huygens, who proved the curve to be tautochronous.

00His enquiries into evolutes enabled him to prove that the evolute of a cycloid was an equal cycloid, and by utilizing this property he constructed the isochronal pendulum generally known as the cycloidal pendulum.

00In 1697 John Bernoulli proposed the famous problem of the brachistochrone (see Mechanics), and it was proved by Leibnitz, Newton and several others that the cycloid was the required curve.

00The method by which the cycloid is generated shows that it consists of an infinite number of cusps placed along the fixed line and separated by a constant distance equal to the circumference of the rolling circle.

00The name cycloid is usually restricted to the portion between two consecutive cusps (fig.

00The b co-ordinates of any point R on the a ?/' t11®V1 a cycloid are expressible in the form x=a(8-}-sin 0); y=a (I -cos 0), M where the co-ordinate axes are the tangent at the vertex 0 and the axis of the curve, a is the radius of the generating circle, and 0 the angle R'CO, where RR' is parallel to LM and C is the centre of the circle in its symmetric position.

00The intrinsic P equation is s =4a sin 4,, and the equation to the evolute is s= 4a cos 1P, which proves the evolute to be a similar cycloid placed as in fig.

00the original cycloid.

00van Schooten, and more completely by John Wallis, who showed that they possessed properties similar to those of the common cycloid.

001, curve c. The companion to the cycloid is a curve so named on account of its similarity of construction, form and equation to the common cycloid.

00Draw any line DE perpendicular to AB and meeting the circle in E, and take a point P on DE such that the line DP =arc BE; then the locus of P is the companion to the cycloid.

00He announced that the curve required in the first problem must be a cycloid, and he gave a method of determining it.

00Among these subjects were the transit of Mercury, the Aurora Borealis, the figure of the earth, the observation of the fixed stars, the inequalities in terrestrial gravitation, the application of mathematics to the theory of the telescope, the limits of certainty in astronomical observations, the solid of greatest attraction, the cycloid, the logistic curve, the theory of comets, the tides, the law of continuity, the double refraction micrometer, various problems of spherical trigonometry, &c. In 1742 he was consulted, with other men of science, by the pope, Benedict XIV., as to the best means of securing the stability of the dome of St Peter's, Rome, in which a crack had been discovered.

00The publication amongst Torricelli's Opera geometrica (Florence, 1644) of a tract on the properties of the cycloid involved him in a controversy with G.

00He 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.

00The determination of the true relation between the length of a pendulum and the time of its oscillation; the invention of the theory of evolutes; the discovery, hence ensuing, that the cycloid is its own evolute, and is strictly isochronous; the ingenious although practically inoperative idea of correcting the "circular error" of the pendulum by applying cycloidal cheeks to clocks - were all contained in this remarkable treatise.

01- Burrowing snakes, mostly small, which have the body covered with smooth, shiny, uniform cycloid scales The teeth are restricted to the small maxillary bones.

01The vermiform body is covered with cycloid imbricating scales, devoid of osteoderms. Limbs and even their arches are absent, excepting a pair of flaps which represent the hind-limbs in the males.

01Many fish inhabited the Carboniferous seas and most of these were Elasmobranchs, sharks with crushing pavement teeth (Psammodus), adapted for grinding the shells of brachiopods, crustaceans, &c. Other sharks had piercing teeth (Cladoselache and Cladodus); some, the petalodonts, had peculiar cycloid cutting teeth.

01

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