Accordingly, 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.'
The 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.
Pascal 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.
There 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.
It 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.
In 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.
CYCLOID (from Gr.
The 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.
The cycloid is the simplest member of the class of curves known as roulettes.
No mention of the cycloid has been found in writings prior to the 15th century.
Evangelista 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.
He there shows that the cycloid was investigated by Carolus Bovillus about r 500, and by Cardinal Cusanus (Nicolaus de Cusa) as early as 1451.
Many other mathematicians have written on the cycloid - Blaise Pascal, W.
A famous period in the history of the cycloid is marked by a bitter controversy which sprang up between Descartes and Roberval.
Among other early writers on the cycloid were Phillippe de Lahire (1640-1718) and Francois Nicole (1683-1758).
The mechanical properties of the cycloid were investigated by Christiaan Huygens, who proved the curve to be tautochronous.
His 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.
The 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.
The 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.
1, 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.
He announced that the curve required in the first problem must be a cycloid, and he gave a method of determining it.
Among 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.
The publication amongst Torricelli's Opera geometrica (Florence, 1644) of a tract on the properties of the cycloid involved him in a controversy with G.
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.
The 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.
- Burrowing snakes, mostly small, which have the body covered with smooth, shiny, uniform cycloid scales The teeth are restricted to the small maxillary bones.
The 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.
Many 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.
The two treatises on the cycloid and on the cissoid, &c., and the Mechanica contain many results which were then new and valuable.