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96-777: The catenary suspension bridge of Real Ferdinando or the Ferdinandeo Bridge over the River Garigliano was the first iron catenary suspension bridge built in Italy , and one of the earliest in continental Europe. This bridge, which was technologically advanced for its age, was built in 1832 by the Bourbon Kingdom of Two Sicilies The engineer who designed the bridge was Luigi Giura . 41°14′32″N 13°46′20″E  /  41.24222°N 13.77222°E  / 41.24222; 13.77222 Catenary The catenary curve has

192-462: A , {\displaystyle {\frac {d\varphi }{ds}}={\frac {\cos ^{2}\varphi }{a}},} and eliminating φ {\displaystyle \varphi } gives the Cesàro equation κ = a s 2 + a 2 , {\displaystyle \kappa ={\frac {a}{s^{2}+a^{2}}},} where κ {\displaystyle \kappa }

288-425: A 2 ( e x a + e − x a ) , {\displaystyle y=a\cosh \left({\frac {x}{a}}\right)={\frac {a}{2}}\left(e^{\frac {x}{a}}+e^{-{\frac {x}{a}}}\right),} where cosh is the hyperbolic cosine function , and where a is the distance of the lowest point above the x axis. All catenary curves are similar to each other, since changing

384-411: A cosh ⁡ x a + b {\displaystyle y=a\cosh {\frac {x}{a}}+b} revolved about the x -axis. In the mathematical model the chain (or cord, cable, rope, string, etc.) is idealized by assuming that it is so thin that it can be regarded as a curve and that it is so flexible any force of tension exerted by the chain is parallel to the chain. The analysis of

480-572: A wave theory of light . His is the first-recorded hypothesis of the cause of the expansion of matter by heat, of air's composition by small particles in constant motion that thus generate its pressure, and of heat as energy. In physics, Hooke inferred that gravity obeys an inverse square law and arguably was the first to hypothesise such a relation in planetary motion, a principle Isaac Newton furthered and formalised in Newton's law of universal gravitation . Priority over this insight contributed to

576-415: A Latin anagram in an appendix to his Description of Helioscopes, where he wrote that he had found "a true mathematical and mechanical form of all manner of Arches for Building." He did not publish the solution to this anagram in his lifetime, but in 1705 his executor provided it as ut pendet continuum flexile, sic stabit contiguum rigidum inversum , meaning "As hangs a flexible cable so, inverted, stand

672-435: A U-like shape, superficially similar in appearance to a parabola , which it is not. The curve appears in the design of certain types of arches and as a cross section of the catenoid —the shape assumed by a soap film bounded by two parallel circular rings. The catenary is also called the alysoid , chainette , or, particularly in the materials sciences, an example of a funicular . Rope statics describes catenaries in

768-548: A balance-controlled watch before the Royal Society, may support Hooke's claim to priority for the idea. Nevertheless, it is Huygens who is credited with building the first watch to use a balance spring. Hooke's announcement of his law of elasticity using an anagram was a method scientists, such as Hooke, Huygens and Galileo , sometimes used to establish priority for a discovery without revealing details. Hooke used mechanical analogues to understand fundamental processes such as

864-591: A brass clock dismantled, he built a wooden replica that "would go". Hooke's father died in October 1648, leaving £40 in his will to Robert (plus another £10 held over from his grandmother). At the age of 13, he took this to London to become an apprentice to the celebrated painter Peter Lely . Hooke also had "some instruction in drawing" from the limner Samuel Cowper but "the smell of the Oil Colours did not agree with his Constitution, increasing his Head-ache to which he

960-523: A bridge: I have lately received from Italy a treatise on the equilibrium of arches, by the Abbé Mascheroni. It appears to be a very scientifical work. I have not yet had time to engage in it; but I find that the conclusions of his demonstrations are, that every part of the catenary is in perfect equilibrium. It is often said that Galileo thought the curve of a hanging chain was parabolic. However, in his Two New Sciences (1638), Galileo wrote that

1056-400: A classic statics problem involving a hanging rope. Mathematically, the catenary curve is the graph of the hyperbolic cosine function. The surface of revolution of the catenary curve, the catenoid , is a minimal surface , specifically a minimal surface of revolution . A hanging chain will assume a shape of least potential energy which is a catenary. Galileo Galilei in 1638 discussed

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1152-494: A curator to furnish the society with experiments, and this was unanimously passed and Hooke was named on Boyle's recommendation. The Society did not have a reliable income to fully fund the post of Curator of Experiments but in 1664, John Cutler settled an annual gratuity of £50 on the Society to found a "Mechanick" lectureship at Gresham College on the understanding the Society would appoint Hooke to this task. On 27 June 1664, Hooke

1248-421: A disgruntled, selfish, anti-social curmudgeon. For example, Arthur Berry said Hooke "claimed credit for most of the scientific discoveries of the time". Sullivan wrote he was "positively unscrupulous" and had an "uneasy apprehensive vanity" in dealings with Newton. Manuel described Hooke as "cantankerous, envious, vengeful". According to More, Hooke had both a "cynical temperament" and a "caustic tongue". Andrade

1344-485: A dog could be kept alive with its thorax opened, provided air was pumped in and out of its lungs. He noted the difference between venous and arterial blood, and thus demonstrated that the Pabulum vitae ("food of life") and flammae [flames] were the same thing. There were also experiments on gravity, the falling of objects, the weighing of bodies, the measurement of barometric pressure at different heights, and

1440-420: A hanging cord is only an approximate parabola, correctly observing that this approximation improves in accuracy as the curvature gets smaller and is almost exact when the elevation is less than 45°. The fact that the curve followed by a chain is not a parabola was proven by Joachim Jungius (1587–1657); this result was published posthumously in 1669. The application of the catenary to the construction of arches

1536-580: A mechanism that improved the regularity of pendulum clocks used for astronomical time-keeping. Hooke characterised his Oxford days as the foundation of his lifelong passion for science. The friends he made there, particularly Christopher Wren , were important to him throughout his career. Willis introduced Hooke to Robert Boyle , who the Club sought to attract to Oxford. In 1655, Boyle moved to Oxford and Hooke became nominally his assistant but in practice his co-experimenter. Boyle had been working on gas pressures;

1632-612: A medium for transmitting attraction and repulsion between separated celestial bodies, Hooke argued for an attracting principle of gravitation in Micrographia (1665). In a communication to the Royal Society in 1666, he wrote: I will explain a system of the world very different from any yet received. It is founded on the following positions. 1. That all the heavenly bodies have not only a gravitation of their parts to their own proper centre, but that they also mutually attract each other within their spheres of action. 2. That all bodies having

1728-513: A new hypothesis from Paris about planetary motions, which he described at length; efforts to carry out or improve national surveys; and the difference of latitude between London and Cambridge. Newton's reply offered "a fansy of my own" about a terrestrial experiment rather than a proposal about celestial motions that might detect the Earth's motion; the experiment would use a body suspended in air and then dropped. Hooke wanted to discern how Newton thought

1824-432: A particularly keen eye and was an adept mathematician, neither of which applied to Boyle. Hooke taught Boyle Euclid's Elements and Descartes 's Principles of Philosophy ; it also caused them to recognise fire as a chemical reaction and not, as Aristotle taught, a fundamental element of nature. According to Henry Robinson, Librarian of The Royal Society in 1935: Without his weekly experiments and prolific work

1920-419: A place at Christ Church , Oxford , receiving free tuition and accommodation as an organist and a chorister , and a basic income as a servitor , despite the fact he did not officially matriculate until 1658. In 1662, Hooke was awarded a Master of Arts degree. While a student at Oxford, Hooke was also employed as an assistant to Dr Thomas Willis  – a physician, chemist and member of

2016-493: A segment using the fact that these forces must be in balance if the chain is in static equilibrium . Let the path followed by the chain be given parametrically by r = ( x , y ) = ( x ( s ), y ( s )) where s represents arc length and r is the position vector . This is the natural parameterization and has the property that d r d s = u {\displaystyle {\frac {d\mathbf {r} }{ds}}=\mathbf {u} } where u

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2112-435: A simple motion, will continue to move in a straight line, unless continually deflected from it by some extraneous force, causing them to describe a circle, an ellipse, or some other curve. 3. That this attraction is so much the greater as the bodies are nearer. As to the proportion in which those forces diminish by an increase of distance, I own I have not discovered it. ... Hooke's 1674 Gresham lecture, An Attempt to Prove

2208-444: A special case of marine vehicles moving although moored by the two catenaries each of one or more cables (wire ropes or chains) passing through the vehicle and moved along by motorized sheaves. The catenaries can be evaluated graphically. The equation of a catenary in Cartesian coordinates has the form y = a cosh ⁡ ( x a ) =

2304-620: A spinal deformity that was consistent with a diagnosis of Scheuermann's kyphosis , giving him in middle and later years a "thin and crooked body, over-large head and protruding eyes". Approaching these in a scientific spirit, he experimented with self-medication, diligently recording symptoms, substances and effects in his diary. He regularly used sal ammoniac , emetics, laxatives and opiates, which appear to have had an increasing effect on his physical and mental health over time. Hooke died in London on 3 March 1703, having been blind and bedridden during

2400-399: A triangle, but the catenary must have parameters corresponding to the shape and dimensions of the wheels. Over any horizontal interval, the ratio of the area under the catenary to its length equals a , independent of the interval selected. The catenary is the only plane curve other than a horizontal line with this property. Also, the geometric centroid of the area under a stretch of catenary

2496-399: Is a unit tangent vector . A differential equation for the curve may be derived as follows. Let c be the lowest point on the chain, called the vertex of the catenary. The slope ⁠ dy / dx ⁠ of the curve is zero at c since it is a minimum point. Assume r is to the right of c since the other case is implied by symmetry. The forces acting on the section of

2592-588: Is attributed to Robert Hooke , whose "true mathematical and mechanical form" in the context of the rebuilding of St Paul's Cathedral alluded to a catenary. Some much older arches approximate catenaries, an example of which is the Arch of Taq-i Kisra in Ctesiphon . In 1671, Hooke announced to the Royal Society that he had solved the problem of the optimal shape of an arch, and in 1675 published an encrypted solution as

2688-422: Is convenient to write a = T 0 w {\displaystyle a={\frac {T_{0}}{w}}} which is the length of chain whose weight is equal in magnitude to the tension at c . Then d y d x = s a {\displaystyle {\frac {dy}{dx}}={\frac {s}{a}}} is an equation defining the curve. The horizontal component of

2784-481: Is credited as one of the first scientists to investigate living things at microscopic scale in 1665, using a compound microscope that he designed. Hooke was an impoverished scientific inquirer in young adulthood who went on to become one of the most important scientists of his time. After the Great Fire of London in 1666, Hooke (as a surveyor and architect) attained wealth and esteem by performing more than half of

2880-494: Is known of Hooke's early life comes from an autobiography he commenced in 1696 but never completed; Richard Waller FRS mentions it in his introduction to The Posthumous Works of Robert Hooke, M.D. S.R.S. , which was printed in 1705. The work of Waller, along with John Ward 's Lives of the Gresham Professors , and John Aubrey 's Brief Lives form the major near-contemporaneous biographical accounts of his life. Hooke

2976-415: Is the curvature . The radius of curvature is then ρ = a sec 2 ⁡ φ , {\displaystyle \rho =a\sec ^{2}\varphi ,} which is the length of the normal between the curve and the x -axis. When a parabola is rolled along a straight line, the roulette curve traced by its focus is a catenary. The envelope of the directrix of

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3072-473: Is the ideal shape for a freestanding arch of constant thickness, the Gateway Arch is narrower near the top. According to the U.S. National Historic Landmark nomination for the arch, it is a " weighted catenary " instead. Its shape corresponds to the shape that a weighted chain, having lighter links in the middle, would form. In free-hanging chains, the force exerted is uniform with respect to length of

3168-638: Is the length of the segment of chain between c and r . The chain is in equilibrium so the sum of three forces is 0 , therefore T cos ⁡ φ = T 0 {\displaystyle T\cos \varphi =T_{0}} and T sin ⁡ φ = w s , {\displaystyle T\sin \varphi =ws\,,} and dividing these gives d y d x = tan ⁡ φ = w s T 0 . {\displaystyle {\frac {dy}{dx}}=\tan \varphi ={\frac {ws}{T_{0}}}\,.} It

3264-405: Is the midpoint of the perpendicular segment connecting the centroid of the curve itself and the x -axis. A moving charge in a uniform electric field travels along a catenary (which tends to a parabola if the charge velocity is much less than the speed of light c ). The surface of revolution with fixed radii at either end that has minimum surface area is a catenary y =

3360-444: Is transferred to a form which is then used as a guide for the placement of bricks or other building material. The Gateway Arch in St. Louis, Missouri , United States is sometimes said to be an (inverted) catenary, but this is incorrect. It is close to a more general curve called a flattened catenary, with equation y = A  cosh( Bx ) , which is a catenary if AB = 1 . While a catenary

3456-565: Is usually painted of Hooke as a morose ... recluse is completely false". He interacted with noted artisans such as clock-maker Thomas Tompion and instrument-maker Christopher Cocks (Cox). Hooke often met Christopher Wren, with whom he shared many interests, and had a lasting friendship with John Aubrey. His diaries also make frequent reference to meetings at coffeehouses and taverns, as well as to dinners with Robert Boyle. On many occasions, Hooke took tea with his lab assistant Harry Hunt. Although he largely lived alone – apart from

3552-572: The Great Red Spot of Jupiter for two hours as it moved across the planet's face. In March 1665, he published his findings and from them, the Italian astronomer Giovanni Cassini calculated the rotation period of Jupiter to be nine hours and fifty-five minutes. One of the most-challenging problems Hooke investigated was the measurement of the distance from Earth to a star other than the Sun. Hooke selected

3648-608: The Oxford Philosophical Club . The Philosophical Club had been founded by John Wilkins , Warden of Wadham College , who led this important group of scientists who went on to form the nucleus of the Royal Society . In 1659, Hooke described to the Club some elements of a method of heavier-than-air flight but concluded human muscles were insufficient to the task. Through the Club, Hooke met Seth Ward (the University's Savilian Professor of Astronomy ) and developed for Ward

3744-413: The balance spring or hairspring, which for the first time enabled a portable timepiece – a watch – to keep time with reasonable accuracy. A bitter dispute between Hooke and Christiaan Huygens on the priority of this invention was to continue for centuries after the death of both but a note dated 23 June 1670 in the journals of the Royal Society, describing a demonstration of

3840-494: The property line surveys and assisting with the city's rapid reconstruction. Often vilified by writers in the centuries after his death, his reputation was restored at the end of the twentieth century and he has been called "England's Leonardo [da Vinci] ". Hooke was a Fellow of the Royal Society and from 1662, he was its first Curator of Experiments. From 1665 to 1703, he was also Professor of Geometry at Gresham College . Hooke began his scientific career as an assistant to

3936-598: The "notion" of "the rule of the decrease of Gravity, being reciprocally as the squares of the distances from the Center". At the same time, according to Edmond Halley 's contemporaneous report, Hooke agreed "the Demonstration of the Curves generated thereby" was wholly Newton's. According to a 2002 assessment of the early history of the inverse square law: "by the late 1660s, the assumption of an 'inverse proportion between gravity and

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4032-646: The Attraction always is in a duplicate proportion to the Distance from the Center Reciprocall, and Consequently that the Velocity will be in a subduplicate proportion to the Attraction and Consequently as Kepler Supposes Reciprocall to the Distance". (Hooke's inference about the velocity is incorrect. ) In 1686, when the first book of Newton's Principia was presented to the Royal Society, Hooke said he had given Newton

4128-473: The Dutch scientist Antonie van Leeuwenhoek went on to develop increased magnification and so reveal protozoa , blood cells and spermatozoa . Micrographia also contains Hooke's, or perhaps Boyle's and Hooke's, ideas on combustion. Hooke's experiments led him to conclude combustion involves a component of air, a statement with which modern scientists would agree but that was not understood widely, if at all, in

4224-698: The Motion of the Earth by Observations (published 1679), said gravitation applies to "all celestial bodies" and restated these three propositions. Hooke's statements up to 1674 make no mention, however, that an inverse square law applies or might apply to these attractions. His model of gravitation was also not yet universal, though it approached universality more closely than previous hypotheses. Hooke did not provide accompanying evidence or mathematical demonstration; he stated in 1674: "Now what these several degrees [of gravitational attraction] are I have not yet experimentally verified", indicating he did not yet know what law

4320-427: The Royal Society's correspondence; Hooke therefore wanted to hear from members about their research or their views about the research of others. Hooke asked Newton's opinions about various matters. Among other items, Hooke mentioned "compounding the celestial motions of the planets of a direct motion by the tangent and an attractive motion towards the central body"; his "hypothesis of the lawes or causes of springinesse";

4416-509: The Society could scarcely have survived, or, at least, would have developed in a quite different way. It is scarcely an exaggeration to say that he was, historically, the creator of the Royal Society. The Royal Society for the Improvement of Natural Knowledge by Experiment was founded in 1660 and given its Royal Charter in July 1662. On 5 November 1661, Robert Moray proposed the appointment of

4512-449: The catenary in the book Two New Sciences recognizing that it was different from a parabola . The mathematical properties of the catenary curve were studied by Robert Hooke in the 1670s, and its equation was derived by Leibniz , Huygens and Johann Bernoulli in 1691. Catenaries and related curves are used in architecture and engineering (e.g., in the design of bridges and arches so that forces do not result in bending moments). In

4608-406: The catenary is the curve which, when rotated about the x -axis, gives the surface of minimum surface area (the catenoid ) for the given bounding circles. Nicolas Fuss gave equations describing the equilibrium of a chain under any force in 1796. Catenary arches are often used in the construction of kilns . To create the desired curve, the shape of a hanging chain of the desired dimensions

4704-404: The chain from c to r are the tension of the chain at c , the tension of the chain at r , and the weight of the chain. The tension at c is tangent to the curve at c and is therefore horizontal without any vertical component and it pulls the section to the left so it may be written (− T 0 , 0) where T 0 is the magnitude of the force. The tension at r is parallel to

4800-419: The chain, and so the chain follows the catenary curve. The same is true of a simple suspension bridge or "catenary bridge," where the roadway follows the cable. A stressed ribbon bridge is a more sophisticated structure with the same catenary shape. However, in a suspension bridge with a suspended roadway, the chains or cables support the weight of the bridge, and so do not hang freely. In most cases

4896-410: The curve at r and pulls the section to the right. The tension at r can be split into two components so it may be written T u = ( T cos φ , T sin φ ) , where T is the magnitude of the force and φ is the angle between the curve at r and the x -axis (see tangential angle ). Finally, the weight of the chain is represented by (0, − ws ) where w is the weight per unit length and s

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4992-402: The curve for an optimal arch is similar except that the forces of tension become forces of compression and everything is inverted. An underlying principle is that the chain may be considered a rigid body once it has attained equilibrium. Equations which define the shape of the curve and the tension of the chain at each point may be derived by a careful inspection of the various forces acting on

5088-440: The deaths of both Newton and Hooke, Alexis Clairaut , mathematical astronomer eminent in his own right in the field of gravitational studies, reviewed Hooke's published work on gravitation. According to Stephen Peter Rigaud , Clairaut wrote: "The example of Hooke and that of Kepler [serves] to show what a distance there is between a truth that is glimpsed and a truth that is demonstrated". I. Bernard Cohen said: "Hooke's claim to

5184-505: The diversion he gave me from my other studies to think on these things & for his dogmaticalness in writing as if he had found the motion in the Ellipsis, which inclined me to try it. Whilst Newton was primarily a pioneer in mathematical analysis and its applications, and optical experimentation, Hooke was a creative experimenter of such great range who left some of his ideas, such as those about gravitation, undeveloped. In 1759, decades after

5280-737: The earliest-recorded observation of a microorganism, the microfungus Mucor . Hooke coined the term " cell ", suggesting a resemblance between plant structures and honeycomb cells. The hand-crafted, leather-and-gold-tooled microscope he designed and used to make the observations for Micrographia , which Christopher Cock made for him in London, is on display at the National Museum of Health and Medicine in Maryland . Hooke's work developed from that of Henry Power , who published his microscopy work in Experimental Philosophy (1663); in turn,

5376-482: The falling body could experimentally reveal the Earth's motion by its direction of deviation from the vertical but Hooke went on hypothetically to consider how its motion could continue if the solid Earth had not been in the way, on a spiral path to the centre. Hooke disagreed with Newton's idea of the body's continuing motion. A further short correspondence developed; towards the end of it, writing on 6 January 1680 to Newton, Hooke communicated his "supposition ... that

5472-465: The first practical Gregorian telescope that used a silvered glass mirror. In 1660, Hooke discovered the law of elasticity that bears his name and describes the linear variation of tension with extension in an elastic spring. Hooke first described this discovery in an anagram "ceiiinosssttuv", whose solution he published in 1678 as Ut tensio, sic vis ("As the extension, so the force"). His work on elasticity culminated in his development of

5568-471: The formation of these craters and concluded their existence meant the Moon must have its own gravity, a radical departure from the contemporaneous Aristotelian celestial model . He also was an early observer of the rings of Saturn , and discovered one of the first-observed double-star systems Gamma Arietis in 1664. To achieve these discoveries, Hooke needed better instruments than those that were available at

5664-446: The gravitation might follow; and about his whole proposal, he said: "This I only hint at present ... having my self many other things in hand which I would first compleat, and therefore cannot so well attend it" (i.e. "prosecuting this Inquiry"). In November 1679, Hooke initiated a notable exchange of letters with Newton that was published in 1960. Hooke's ostensible purpose was to tell Newton he (Hooke) had been appointed to manage

5760-453: The history of life on Earth and, despite the objections of contemporary naturalists like John Ray  – who found the concept of extinction theologically unacceptable – that in some cases they might represent species that had become extinct through some geological disaster. In a series of lectures in 1668, Hooke proposed the then-heretical idea the Earth's surface had been formed by volcanoes and earthquakes, and that

5856-458: The hyperbolic cosine and sine functions are basic solutions to Maxwell's equations. The symmetric modes consisting of two evanescent waves would form a catenary shape. The word "catenary" is derived from the Latin word catēna , which means " chain ". The English word "catenary" is usually attributed to Thomas Jefferson , who wrote in a letter to Thomas Paine on the construction of an arch for

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5952-428: The invention was, by Hooke's death, in constant use among clock makers. Hooke announced he conceived a way to build a marine chronometer to determine longitude. and with the help of Boyle and others, he attempted to patent it. In the process, Hooke demonstrated a pocket-watch of his own devising that was fitted with a coil spring attached to the arbour of the balance. Hooke's refusal to accept an escape clause in

6048-472: The inverse-square law has masked Newton's far more fundamental debt to him, the analysis of curvilinear orbital motion. In asking for too much credit, Hooke effectively denied to himself the credit due him for a seminal idea". Hooke made important contributions to the science of timekeeping and was intimately involved in the advances of his time; these included refinement of the pendulum as a better regulator for clocks, increased precision of clock mechanisms and

6144-425: The last year of his life. A chest containing £8,000 in money and gold was found in his room at Gresham College . His library contained over 3,000 books in Latin, French, Italian and English. Although he had talked of leaving a generous bequest to the Royal Society, which would have given his name to a library, laboratory and lectures, no will was found and the money passed to a cousin named Elizabeth Stephens. Hooke

6240-522: The latter were responsible for shell fossils being found far above sea level. In 1835, Charles Lyell , the Scottish geologist and associate of Charles Darwin , wrote of Hooke in Principles of Geology : "His treatise ... is the most philosophical production of that age, in regard to the causes of former changes in the organic and inorganic kingdoms of nature". Hooke's scientific model of human memory

6336-497: The love of his life, and he was devastated when she died in 1687. Inwood also mentions "The age difference between him and Grace was commonplace and would not have upset his contemporaries as it does us". The incestous relationship would nevertheless have been frowned upon and tried by an ecclesiastical court had it been discovered, it was not however a capital felony after 1660. Since childhood, Hooke suffered from migraine , tinnitus , dizziness and bouts of insomnia ; he also had

6432-507: The most ingenious book that ever I read in my life". One of the observations in Micrographia is of fossil wood , the microscopic structure of which Hooke compared to that of ordinary wood. This led him to conclude that fossilised objects like petrified wood and fossil shells such as ammonites were the remains of living things that had been soaked in mineral-laden petrifying water. He believed that such fossils provided reliable clues about

6528-556: The motion of a spherical pendulum and of a ball in a hollow cone, to demonstrate central force due to gravity, and a hanging chain net with point loads to provide the optimum shape for a dome with heavy cross on top. Despite continuing reports to the contrary, Hooke did not influence Thomas Newcomen 's invention of the steam engine ; this myth, which originated in an article in the third edition of " Encyclopædia Britannica ", has been found to be mistaken. While many of Hooke's contemporaries, such as Isaac Newton, believed in aether as

6624-440: The movement of pendulums up to 200 ft long (61 m). His biographer Margaret 'Espinasse described him as England's first meteorologist , in her description of his essay Method for making a history of the weather . (Hooke specifies that a thermometer, a hygrometer , a wind gauge and a record sheet be used for proper weather records. ) In May 1664, using a 12 ft (3.7 m) refracting telescope , Hooke observed

6720-406: The offshore oil and gas industry, "catenary" refers to a steel catenary riser , a pipeline suspended between a production platform and the seabed that adopts an approximate catenary shape. In the rail industry it refers to the overhead wiring that transfers power to trains. (This often supports a contact wire, in which case it does not follow a true catenary curve.) In optics and electromagnetics,

6816-450: The parabola is also a catenary. The involute from the vertex, that is the roulette traced by a point starting at the vertex when a line is rolled on a catenary, is the tractrix . Another roulette, formed by rolling a line on a catenary, is another line. This implies that square wheels can roll perfectly smoothly on a road made of a series of bumps in the shape of an inverted catenary curve. The wheels can be any regular polygon except

6912-493: The parameter a is equivalent to a uniform scaling of the curve. The Whewell equation for the catenary is tan ⁡ φ = s a , {\displaystyle \tan \varphi ={\frac {s}{a}},} where φ {\displaystyle \varphi } is the tangential angle and s the arc length . Differentiating gives d φ d s = cos 2 ⁡ φ

7008-488: The physical scientist Robert Boyle . Hooke built the vacuum pumps that were used in Boyle's experiments on gas law and also conducted experiments. In 1664, Hooke identified the rotations of Mars and Jupiter . Hooke's 1665 book Micrographia , in which he coined the term cell , encouraged microscopic investigations. Investigating optics  – specifically light refraction  – Hooke inferred

7104-406: The possibility a vacuum might exist despite Aristotle 's maxim " Nature abhors a vacuum " had just begun to be considered . Hooke developed an air pump for Boyle's experiments rather than use Ralph Greatorex 's pump, which Hooke considered as "too gross to perform any great matter". Hooke's engine enabled the development of the eponymous law that was subsequently attributed to Boyle; Hooke had

7200-688: The proposed exclusive contract for the use of this idea resulted in its abandonment. Hooke developed the principle of the balance spring independently of Huygens and at least five years beforehand. Huygens published his own work in Journal de Scavans in February 1675 and built the first functioning watch to use a balance spring. In 1663 and 1664, Hooke made his microscopic, and some astronomic, observations, which he collated in Micrographia in 1665. His book, which describes observations with microscopes and telescopes, as well as original work in biology, contains

7296-466: The rivalry between Hooke and Newton. In geology and palaeontology , Hooke originated the theory of a terraqueous globe, thus disputing the Biblical view of the Earth's age; he also hypothesised the extinction of species, and argued hills and mountains had become elevated by geological processes. By identifying fossils of extinct species, Hooke presaged the theory of biological evolution . Much of what

7392-672: The roadway is flat, so when the weight of the cable is negligible compared with the weight being supported, the force exerted is uniform with respect to horizontal distance, and the result is a parabola , as discussed below (although the term "catenary" is often still used, in an informal sense). If the cable is heavy then the resulting curve is between a catenary and a parabola. The catenary produced by gravity provides an advantage to heavy anchor rodes . An anchor rode (or anchor line) usually consists of chain or cable or both. Anchor rodes are used by ships, oil rigs, docks, floating wind turbines , and other marine equipment which must be anchored to

7488-542: The seabed. When the rope is slack, the catenary curve presents a lower angle of pull on the anchor or mooring device than would be the case if it were nearly straight. This enhances the performance of the anchor and raises the level of force it will resist before dragging. To maintain the catenary shape in the presence of wind, a heavy chain is needed, so that only larger ships in deeper water can rely on this effect. Smaller boats also rely on catenary to maintain maximum holding power. Cable ferries and chain boats present

7584-493: The servants who ran his home – his niece Grace Hooke and his cousin Tom Giles lived with him for some years as children. Hooke never married. According to his diary, Hooke had a sexual relationship with his niece Grace, after she had turned 16. Grace was in his custody since the age of 10. He also had sexual relations with several maids and housekeepers. Hooke's biographer Stephen Inwood considers Grace to have been

7680-467: The seventeenth century. He also concluded respiration and combustion involve a specific and limited component of air. According to Partington, if "Hooke had continued his experiments on combustion, it is probable that he would have discovered oxygen". Samuel Pepys wrote of the book in his diary on 21 January 16 ⁠ 64 / 65 ⁠ : "Before I went to bed I sat up till two o’clock in my chamber reading of Mr. Hooke's Microscopicall Observations,

7776-446: The square of distance' was rather common and had been advanced by a number of different people for different reasons". In the 1660s, Newton had shown for planetary motion under a circular assumption, force in the radial direction had an inverse-square relation with distance from the centre. Newton, who in May 1686 was presented with Hooke's claim to priority on the inverse square law, denied he

7872-450: The star Gamma Draconis and chose the method of parallax determination. In 1669, after several months of observing, Hooke believed the desired result had been achieved. It is now known his equipment was far too imprecise to obtain an accurate measurement. Hooke's Micrographia contains illustrations of the Pleiades star cluster and lunar craters . He conducted experiments to investigate

7968-817: The supposition, could only guess it was approximately valid "at great distances from the centre". Newton did accept and acknowledge, in all editions of the Principia , Hooke and others had separately appreciated the inverse square law in the solar system. Newton acknowledged Wren, Hooke and Halley in this connection in his "Scholium to Proposition 4" in Book   1. In a letter to Halley, Newton also acknowledged his correspondence with Hooke in 1679–1680 had reawakened his dormant interest in astronomical matters but that did not mean, according to Newton, Hooke had told Newton anything new or original. Newton wrote: Yet am I not beholden to him for any light into that business ... but only for

8064-437: The tension, T cos φ = T 0 is constant and the vertical component of the tension, T sin φ = ws is proportional to the length of chain between r and the vertex. Robert Hooke Robert Hooke FRS ( / h ʊ k / ; 18 July 1635 – 3 March 1703) was an English polymath who was active as a physicist ("natural philosopher"), astronomer, geologist, meteorologist and architect. He

8160-455: The time. Accordingly, he invented three new mechanisms: the Hooke joint , a sophisticated universal joint that allowed his instruments to smoothly follow the apparent motion of the observed body; the first clockwork drive to automate the process; and a micrometer screw that allowed him to achieve a precision of ten seconds of arc . Hooke was dissatisfied with refracting telescopes so he built

8256-495: The touching pieces of an arch." In 1691, Gottfried Leibniz , Christiaan Huygens , and Johann Bernoulli derived the equation in response to a challenge by Jakob Bernoulli ; their solutions were published in the Acta Eruditorum for June 1691. David Gregory wrote a treatise on the catenary in 1697 in which he provided an incorrect derivation of the correct differential equation. Leonhard Euler proved in 1744 that

8352-432: The type in which he was involved seem almost to be the rule rather than the exception. And Hooke's reaction to such controversy involving his own discoveries and inventions seems mild in comparison to the behaviour of some of his contemporaries". The publication of Hooke's diary in 1935 revealed previously unknown details about his social and familial relationships. His biographer Margaret 'Espinasse said: "the picture which

8448-415: The use of the balance spring to improve the timekeeping of watches. Galileo had observed the regularity of a pendulum and Huygens first incorporated it in a clock; in 1668, Hooke demonstrated his new device to keep a pendulum swinging regularly in unsteady conditions. His invention of a tooth-cutting machine enabled a substantial improvement in the accuracy and precision of timepieces. Waller reported

8544-520: Was appointed its Joint Secretary. Although John Aubrey described Hooke as a person of "great virtue and goodness". much has been written about the unpleasant side of Hooke's personality. According to his first biographer Richard Waller, Hooke was "in person, but despicable", and "melancholy, mistrustful, and jealous". Waller's comments influenced other writers for more than 200 years such that many books and articles – especially biographies of Isaac Newton  – portray Hooke as

8640-634: Was born in 1635 in Freshwater, Isle of Wight , to Cecily Gyles and the Anglican priest John Hooke, who was the curate of All Saints' Church, Freshwater . Robert was the youngest, by seven years, of four siblings (two boys and two girls); he was frail and not expected to live. Although his father gave him some instruction in English, (Latin) Grammar and Divinity , Robert's education was largely neglected. Left to his own devices, he made little mechanical toys; seeing

8736-467: Was buried at St Helen's Church, Bishopsgate , in the City of London but the precise location of his grave is unknown. Hooke's role at the Royal Society was to demonstrate experiments from his own methods or at the suggestion of members. Among his earliest demonstrations were discussions of the nature of air and the implosion of glass bubbles that had been sealed with enclosed hot air. He also demonstrated that

8832-436: Was confirmed to the office and on 11 January 1665, he was named Curator by Office for life with an annual salary of £80, which consisting of £30 from the Society and Cutler's £50 annuity. In June 1663, Hooke was elected a Fellow of the Royal Society (FRS). On 20 March 1665, he was also appointed Gresham Professor of Geometry . On 13 September 1667, Hooke became acting Secretary of the Society and on 19 December 1677, he

8928-448: Was ever too much subject", and he became a pupil at Westminster School , living with its master Richard Busby . Hooke quickly mastered Latin, Greek and Euclid's Elements ; he also learnt to play the organ and began his lifelong study of mechanics. He remained an accomplished draughtsman, as he was later to demonstrate in his drawings that illustrate the work of Robert Boyle and Hooke's own Micrographia . In 1653, Hooke secured

9024-480: Was more sympathetic but still described Hooke as "difficult", "suspicious" and "irritable". In October 1675, the Council of the Royal Society considered a motion to expel Hooke because of an attack he made on Christiaan Huygens over scientific priority in watch design but it did not pass. According to Hooke's biographer Ellen Drake: if one studies the intellectual milieu of the time, the controversies and rivalries of

9120-646: Was one of the first of its kind. In a 1682 lecture to the Royal Society, Hooke proposed a mechanical analogue model of human memory that bore little resemblance to the mainly philosophical models of earlier writers. This model addressed the components of encoding, memory capacity, repetition, retrieval, and forgetting – some with surprisingly modern accuracy. According to psychology professor Douglas Hintzman, Hooke's model's most-interesting points are that it allows for attention and other top-down influences on encoding; it uses resonance to implement parallel, cue-dependent retrieval; it explains memory for recency; it offers

9216-509: Was to be credited as author of the idea, giving reasons including the citation of prior work by others. Newton also said that, even if he had first heard of the inverse square proportion from Hooke (which Newton said he had not), he would still have some rights to it because of his mathematical developments and demonstrations. These, he said, enabled observations to be relied upon as evidence of its accuracy while according to Newton, Hooke, without mathematical demonstrations and evidence in favour of

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