Writing of Principia Mathematica
Isaac Newton composed Principia Mathematica during 1685 and 1686, it was published in a first edition on 5 July 1687. Regarded as one of the most important works in both the science of physics and in applied mathematics during the Scientific Revolution, the work underlies much of the technological and scientific advances from the Industrial Revolution which it helped to create. Between 1685 and 1686, Newton had a extensive correspondence with John Flamsteed, the astronomer-royal. Many of the letters are lost, but it is clear that Flamsteed was helpful regarding Kepler's definition of Saturn; the publication of Newton's discoveries led to controversies involving English natural philosopher Robert Hooke, Anthony Lucas, mathematical professor at Liege. English astronomer and mathematician Edmond Halley attempted to mediate and get Newton to agree that Hooke deserved some credit for the invention of "the rule for the decrease of gravity being reciprocally as the squares of the distances from the centre".
Halley acknowledged that although Hooke should be credited for the theory, "the demonstration of the curves generated thereby belonged wholly to Newton". In a letter, Newton responded to Halley as follows: "Sir, In order to let you know the case between Mr Hooke and me, I give you an account of what passed between us in our letters, so far as I could remember. I am confident by circumstances, that Sir Chr. Wren knew the duplicate proportion. I intended in this letter to let you understand the case fully; that what he told me of the duplicate proportion was erroneous, that it reached down from hence to the centre of the earth. "That it is not candid to require me now to confess myself, in print ignorant of the duplicate proportion in the heavens. That in my answer to his first letter I refused his correspondence, told him I had laid philosophy aside, sent him, only the experiment of projectiles, in compliment to sweeten my answer, expected to hear no further from him; that by the same reason he concludes me ignorant of the rest of the duplicate proportion, he may as well conclude me ignorant of the rest of that theory I had read before in his books.
That in one of my papers writ, the proportion of the forces of the planets from the sun, reciprocally duplicate of their distances from him, is expressed, the proportion of our gravit to the moon's conatus recedendi a centro terrae is calculated, though not enough. That when Hugenius put out his Horol. Oscill. A copy being presented to me, in my letter of thanks to him I gave those rules in the end thereof a particular commendation for their usefulness in Philosophy, added out of my aforesaid paper an instance of their usefulness, in comparing the forces of the moon from the earth, earth from the sun. Between ten and eleven years ago there was an hypothesis of mine registered in your books, wherein I hinted a cause of gravity towards the earth and planets, with the dependence of the celestial motions thereon, and I hope I shall not be urged to declare, in print, that I understood not the obvious mathematical condition of my own hypothesis. But, grant I received it afterwards from Mr Hooke, yet have I as great a right to it as to the ellipse.
For as Kepler knew the orb to be not circular but oval, guessed it to be elliptical, so Mr Hooke, without knowing what I have found out since his letters to me, can know no more, but that the proportion was duplicate quam proximè at great distances from the centre, only guessed it to be so and guessed amiss in extending that proportion down to the centre, whereas Kepler guessed right at the ellipse. And so, Mr Hooke found less of the proportion than Kepler of the ellipse. "The
Temperature is a physical quantity expressing hot and cold. It is measured with a thermometer calibrated in one or more temperature scales; the most used scales are the Celsius scale, Fahrenheit scale, Kelvin scale. The kelvin is the unit of temperature in the International System of Units, in which temperature is one of the seven fundamental base quantities; the Kelvin scale is used in science and technology. Theoretically, the coldest a system can be is when its temperature is absolute zero, at which point the thermal motion in matter would be zero. However, an actual physical system or object can never attain a temperature of absolute zero. Absolute zero is denoted as 0 K on the Kelvin scale, −273.15 °C on the Celsius scale, −459.67 °F on the Fahrenheit scale. For an ideal gas, temperature is proportional to the average kinetic energy of the random microscopic motions of the constituent microscopic particles. Temperature is important in all fields of natural science, including physics, Earth science and biology, as well as most aspects of daily life.
Many physical processes are affected by temperature, such as physical properties of materials including the phase, solubility, vapor pressure, electrical conductivity rate and extent to which chemical reactions occur the amount and properties of thermal radiation emitted from the surface of an object speed of sound is a function of the square root of the absolute temperature Temperature scales differ in two ways: the point chosen as zero degrees, the magnitudes of incremental units or degrees on the scale. The Celsius scale is used for common temperature measurements in most of the world, it is an empirical scale, developed by a historical progress, which led to its zero point 0 °C being defined by the freezing point of water, additional degrees defined so that 100 °C was the boiling point of water, both at sea-level atmospheric pressure. Because of the 100-degree interval, it was called a centigrade scale. Since the standardization of the kelvin in the International System of Units, it has subsequently been redefined in terms of the equivalent fixing points on the Kelvin scale, so that a temperature increment of one degree Celsius is the same as an increment of one kelvin, though they differ by an additive offset of 273.15.
The United States uses the Fahrenheit scale, on which water freezes at 32 °F and boils at 212 °F at sea-level atmospheric pressure. Many scientific measurements use the Kelvin temperature scale, named in honor of the Scots-Irish physicist who first defined it, it is a absolute temperature scale. Its zero point, 0 K, is defined to coincide with the coldest physically-possible temperature, its degrees are defined through thermodynamics. The temperature of absolute zero occurs at 0 K = −273.15 °C, the freezing point of water at sea-level atmospheric pressure occurs at 273.15 K = 0 °C. The International System of Units defines a scale and unit for the kelvin or thermodynamic temperature by using the reliably reproducible temperature of the triple point of water as a second reference point; the triple point is a singular state with its own unique and invariant temperature and pressure, along with, for a fixed mass of water in a vessel of fixed volume, an autonomically and stably self-determining partition into three mutually contacting phases, vapour and solid, dynamically depending only on the total internal energy of the mass of water.
For historical reasons, the triple point temperature of water is fixed at 273.16 units of the measurement increment. There is a variety of kinds of temperature scale, it may be convenient to classify them theoretically based. Empirical temperature scales are older, while theoretically based scales arose in the middle of the nineteenth century. Empirically based temperature scales rely directly on measurements of simple physical properties of materials. For example, the length of a column of mercury, confined in a glass-walled capillary tube, is dependent on temperature, is the basis of the useful mercury-in-glass thermometer; such scales are valid only within convenient ranges of temperature. For example, above the boiling point of mercury, a mercury-in-glass thermometer is impracticable. Most materials expand with temperature increase, but some materials, such as water, contract with temperature increase over some specific range, they are hardly useful as thermometric materials. A material is of no use as a thermometer near one of its phase-change temperatures, for example its boiling-point.
In spite of these restrictions, most used practical thermometers are of the empirically based kind. It was used for calorimetry, which contributed to the discovery of thermodynamics. Empirical thermometry has serious drawbacks when judged as a basis for theoretical physics. Empirically based thermometers, beyond their base as simple direct measurements of ordinary physical properties of thermometric materials, can be re-calibrated, by use of theoretical physical reasoning, this can extend their range of adequacy. Theoretically-based temperature scales are based directly on theoretical arguments those of thermodynamics, kinetic theory and quantum mechanics, they rely on theoretical properties of idealized materials. They are more or less comparable with feasible physical devices and materials. Theoretically based temperature scales are used to provide calibrating standards for practi
Standing on the shoulders of giants
The metaphor of dwarfs standing on the shoulders of giants expresses the meaning of "discovering truth by building on previous discoveries". This concept has been traced to the 12th century, attributed to Bernard of Chartres, its most familiar expression in English is by Isaac Newton in 1675: "If I have seen further it is by standing on the shoulders of Giants." The attribution to Bernard of Chartres is due to John of Salisbury. In 1159, John wrote in his Metalogicon: "Bernard of Chartres used to compare us to dwarfs perched on the shoulders of giants, he pointed out that we see more and farther than our predecessors, not because we have keener vision or greater height, but because we are lifted up and borne aloft on their gigantic stature." According to medieval historian Richard William Southern, Bernard was comparing contemporary 12th century scholars to the ancient scholars of Greece and Rome: sums up the quality of the cathedral schools in the history of learning, indeed characterizes the age which opened with Gerbert and Fulbert and closed in the first quarter of the 12th century with Peter Abelard. is not a great claim.
It is a shrewd and just remark, the important and original point was the dwarf could see a little further than the giant. That this was possible was above all due to the cathedral schools with their lack of a well-rooted tradition and their freedom from a defined routine of study; the visual image appears in the stained glass of the south transept of Chartres Cathedral. The tall windows under the Rose Window show the four major prophets of the Hebrew Bible as gigantic figures, the four New Testament evangelists as ordinary-size people sitting on their shoulders; the evangelists, though smaller, "see more" than the huge prophets. The phrase appears in the works of the Jewish tosaphist Isaiah di Trani: Should Joshua the son of Nun endorse a mistaken position, I would reject it out of hand, I do not hesitate to express my opinion, regarding such matters in accordance with the modicum of intelligence allotted to me. I was never arrogant claiming "My Wisdom served me well". Instead I applied to myself the parable of the philosophers.
For I heard the following from the philosophers, The wisest of the philosophers was asked: "We admit that our predecessors were wiser than we. At the same time we criticize their comments rejecting them and claiming that the truth rests with us. How is this possible?" The wise philosopher responded: "Who sees further a dwarf or a giant? A giant for his eyes are situated at a higher level than those of the dwarf, but if the dwarf is placed on the shoulders of the giant who sees further?... So too we are dwarfs astride the shoulders of giants. We master their move beyond it. Due to their wisdom we grow wise and are able to say all that we say, but not because we are greater than they. Diego de Estella took up the quote in the 16th century. Robert Burton, in The Anatomy of Melancholy, quotes Stella thus: I say with Didacus Stella, a dwarf standing on the shoulders of a giant may see farther than a giant himself. Editors of Burton misattributed the quote to Lucan. No reference or allusion to the quote is found there.
In the 17th century, George Herbert, in his Jacula Prudentum, wrote "A dwarf on a giant's shoulders sees farther of the two." Isaac Newton remarked in a letter to his rival Robert Hooke dated February 5, 1676 that: What Des-Cartes did was a good step. You have added much several ways, & in taking the colours of thin plates into philosophical consideration. If I have seen further it is by standing on the sholders of Giants; this has been interpreted by a few writers as a sarcastic remark directed at Hooke's appearance. Although Hooke was not of short stature, he was of slight build and had been afflicted from his youth with a severe kyphosis. However, at this time Hooke and Newton were on good terms and had exchanged many letters in tones of mutual regard. Only when Robert Hooke criticized some of Newton's ideas regarding optics, was Newton so offended that he withdrew from public debate; the two men remained enemies until Hooke's death. Samuel Taylor Coleridge, in The Friend, wrote: The dwarf sees farther than the giant, when he has the giant's shoulder to mount on.
Against this notion, Friedrich Nietzsche argues that a dwarf brings the most sublime heights down to his level of understanding. In the section of Thus Spoke Zarathustra entitled "On the Vision and the Riddle", Zarathustra climbs to great heights with a dwarf on his shoulders to show him his greatest thought. Once there however, the dwarf fails to understand the profundity of the vision and Zarathustra reproaches him for "making things too easy on self." If there is to be anything resembling "progress" in the history of philosophy, Nietzsche in "Philosophy in the Tragic Age of the Greeks" writes, it can only come from those rare giants among men, "each giant calling to his brother through the desolate intervals of time," an idea he got from Schopenhauer's work in Der handschriftliche Nachlass. The British two pound coin bears the inscription STANDING ON THE SHOULDERS OF GIANTS on its edge.
The Dulong–Petit law, a thermodynamic law proposed in 1819 by French physicists Pierre Louis Dulong and Alexis Thérèse Petit, states the classical expression for the molar specific heat capacity of certain chemical elements. Experimentally the two scientists had found that the heat capacity per weight for a number of elements was close to a constant value, after it had been multiplied by a number representing the presumed relative atomic weight of the element; these atomic weights had shortly before been suggested by John Dalton and modified by Jacob Berzelius. In modern terms and Petit found that the heat capacity of a mole of many solid elements is about 3R, where R is the modern constant called the universal gas constant. Dulong and Petit were unaware of the relationship with R, since this constant had not yet been defined from the kinetic theory of gases; the value of 3R is about 25 joules per kelvin, Dulong and Petit found that this was the heat capacity of certain solid elements per mole of atoms they contained.
The modern theory of the heat capacity of solids states that it is due to lattice vibrations in the solid and was first derived in crude form from this assumption by Albert Einstein in 1907. The Einstein solid model thus gave for the first time a reason why the Dulong–Petit law should be stated in terms of the classical heat capacities for gases. An equivalent statement of the Dulong–Petit law in modern terms is that, regardless of the nature of the substance, the specific heat capacity c of a solid element is equal to 3R/M, where R is the gas constant and M is the molar mass. Thus, the heat capacity per mole of many elements is 3R; the initial form of the Dulong–Petit law was: c M = K where K is a constant which we know today is about 3R. In modern terms the mass m of the sample divided by molar mass M gives the number of moles n. M / M = n Therefore, using uppercase C for the full heat capacity, we have: C = C / n = K = 3 R or C / n = 3 R. Therefore, the heat capacity of most solid crystalline substances is 3R per mole of substance.
Dulong and Petit did not state their law in terms of the gas constant R. Instead, they measured the values of heat capacities of substances and found them smaller for substances of greater atomic weight as inferred by Dalton and other early atomists. Dulong and Petit found that when multiplied by these atomic weights, the value for the heat capacity per mole was nearly constant, equal to a value, recognized to be 3R. In other modern terminology, the dimensionless heat capacity is equal to 3; the law can be written as a function of the total number of atoms N in the sample: C / N = 3 k B,where kB is Boltzmann constant. Despite its simplicity, Dulong–Petit law offers good prediction for the heat capacity of many elementary solids with simple crystal structure at high temperatures; this agreement is because in the classical statistical theory of Ludwig Boltzmann, the heat capacity of solids approaches a maximum of 3R per mole of atoms because full vibrational-mode degrees of freedom amount to 3 degrees of freedom per atom, each corresponding to a quadratic kinetic energy term and a quadratic potential energy term.
By the equipartition theorem, the average of each quadratic term is 1⁄2 RT per mole. Multiplied by 3 degrees of freedom and the two terms per degree of freedom, this amounts to 3R per mole heat capacity; the Dulong–Petit law fails at room temperatures for light atoms bonded to each other, such as in metallic beryllium and in carbon as diamond. Here, it predicts higher heat capacities than are found, with the difference due to higher-energy vibrational modes not being populated at room temperatures in these substances. In the low temperature region, where the quantum mechanical nature of energy storage in all solids manifests itself with larger and larger effect, the law fails for all substances. For crystals under such conditions, the Debye model, an extension of the Einstein theory that accounts for statistical distributions in atomic vibration when there are lower amounts of energy to distribute, works well. A system of vibrations in a crystalline solid lattice can be modelled as an Einstein solid, i.e.by considering N quantum harmonic oscillator potentials along each degree of freedom.
The free energy of the system can be written as F = N ε 0 + N k B T ∑ α log where the index α sums over all the degrees of freedom. In the 1907 Einstein model we consider only the high-energy limit: k B T ≫ ℏ ω α
Method of Fluxions
Method of Fluxions is a book by Isaac Newton. The book was completed in 1671, published in 1736. Fluxion is Newton's term for a derivative, he developed the method at Woolsthorpe Manor during the closing of Cambridge during the Great Plague of London from 1665 to 1667, but did not choose to make his findings known. Gottfried Leibniz developed his form of calculus independently around 1673, 7 years after Newton had developed the basis for differential calculus, as seen in surviving documents like “the method of fluxions and fluents..." from 1666. Leibniz however published his discovery of differential calculus in 1684, nine years before Newton formally published his fluxion notation form of calculus in part during 1693; the calculus notation in use today is that of Leibniz, although Newton's dot notation for differentiation x ˙ for denoting derivatives with respect to time is still in current use throughout mechanics and circuit analysis. Newton's Method of Fluxions was formally published posthumously, but following Leibniz's publication of the calculus a bitter rivalry erupted between the two mathematicians over who had developed the calculus first and so Newton no longer hid his knowledge of fluxions.
For a period of time encompassing Newton's working life, the discipline of analysis was a subject of controversy in the mathematical community. Although analytic techniques provided solutions to long-standing problems, including problems of quadrature and the finding of tangents, the proofs of these solutions were not known to be reducible to the synthetic rules of Euclidean geometry. Instead, analysts were forced to invoke infinitesimal, or "infinitely small", quantities to justify their algebraic manipulations; some of Newton's mathematical contemporaries, such as Isaac Barrow, were skeptical of such techniques, which had no clear geometric interpretation. Although in his early work Newton used infinitesimals in his derivations without justifying them, he developed something akin to the modern definition of limits in order to justify his work. Method of Fluxions at the Internet Archive
In building and construction, the R-value is a measure of how well a two-dimensional barrier, such as a layer of insulation, a window or a complete wall or ceiling, resists conductive flow of heat. R-values measure the thermal resistance per unit of a barrier's exposed area; the greater the R-value, the greater the resistance, so the better the thermal insulating properties of the barrier. R-values are used in describing effectiveness of insulating material and in analysis of heat flow across assemblies under steady-state conditions. Heat flow through a barrier is driven by temperature difference between two sides of the barrier, the R-value quantifies how the object resists this drive: The temperature difference divided by the R-value and multiplied by the surface area of the barrier gives the total rate of heat flow through the barrier, as measured in watts or in BTUs per hour; as long as the materials involved are dense solids in direct mutual contact, R-values are additive. Note that the R-value is the building industry term for what is in other contexts called "thermal resistance per unit area."
It is sometimes denoted RSI-value. An R-value can be given for an assembly of materials. In the case of materials, it is expressed in terms of R-value per unit length; the latter can be misleading in the case of low-density building thermal insulations, for which R-values are not additive: their R-value per inch is not constant as the material gets thicker, but rather decreases. The units of an R-value are not explicitly stated, so it is important to decide from context which units are being used: an R-value expressed in I-P units is about 5.68 times larger than when expressed in SI units, so that, for example, a window, R-2 in I-P units has an RSI of 0.35. For R-values there is no difference between imperial units; as far as how R-values are reported, all of the following mean the same thing: "this is an R-2 window". The more a material is intrinsically able to conduct heat, as given by its thermal conductivity, the lower its R-value. On the other hand, the thicker the material, the higher its R-value.
Sometimes heat transfer processes other than conduction contribute to heat transfer within the material. In such cases, it is useful to introduce an "apparent thermal conductivity", which captures the effects of all three kinds of processes, to define the R-value in general as R = thickness of the specimen apparent thermal conductivity; this comes at a price, however: R-values that include non-conductive processes may no longer be additive and may have significant temperature dependence. In particular, for a loose or porous material, the R-value per inch depends on the thickness always so that it decreases with increasing thickness. For similar reasons, the R-value per inch depends on the temperature of the material increasing with decreasing temperature. In construction it is common to treat R-values as independent of temperature. Note that an R-value may not account for radiative or convective processes at the material's surface, which may be an important factor for some applications; the R-value is the reciprocal of the thermal transmittance of a assembly.
The U. S. construction industry prefers to use R-values, because they are additive and because bigger values mean better insulation, neither of, true for U-factors. The U-factor or U-value is the overall heat transfer coefficient that describes how well a building element conducts heat or the rate of transfer of heat through one square metre of a structure divided by the difference in temperature across the structure; the elements are assemblies of many layers of components such as those that make up walls/floors/roofs etc. It measures the rate of heat transfer through a building element over a given area under standardised conditions; the usual standard is at 50 % humidity with no wind. It is expressed in watts per meter squared kelvin; this means. A low U-value indicates high levels of insulation, they are useful as it is a way of predicting the composite behavior of an entire building element rather than relying on the properties of individual materials. In most countries the properties of specific materials are indicated by the thermal conductivity, sometimes called a k-value or lambda-value.
The thermal conductivity is the ability of a material to conduct heat. Expanded polystyrene has a k-value of around 0.033 W/m
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between those layers. Turbulence is observed in everyday phenomena such as surf, fast flowing rivers, billowing storm clouds, or smoke from a chimney, most fluid flows occurring in nature or created in engineering applications are turbulent. Turbulence is caused by excessive kinetic energy in parts of a fluid flow, which overcomes the damping effect of the fluid's viscosity. For this reason turbulence is realized in low viscosity fluids. In general terms, in turbulent flow, unsteady vortices appear of many sizes which interact with each other drag due to friction effects increases; this increases the energy needed to pump fluid through a pipe. Turbulence can be exploited, for example, by devices such as aerodynamic spoilers on aircraft that "spoil" the laminar flow to increase drag and reduce lift.
The onset of turbulence can be predicted by the dimensionless Reynolds number, the ratio of kinetic energy to viscous damping in a fluid flow. However, turbulence has long resisted detailed physical analysis, the interactions within turbulence create a complex phenomenon. Richard Feynman has described turbulence as the most important unsolved problem in classical physics. Smoke rising from a cigarette. For the first few centimeters, the smoke is laminar; the smoke plume becomes turbulent as its Reynolds number increases with increases in flow velocity and characteristic lengthscale. Flow over a golf ball. If the golf ball were smooth, the boundary layer flow over the front of the sphere would be laminar at typical conditions. However, the boundary layer would separate early, as the pressure gradient switched from favorable to unfavorable, creating a large region of low pressure behind the ball that creates high form drag. To prevent this, the surface is dimpled to promote turbulence; this results in higher skin friction, but it moves the point of boundary layer separation further along, resulting in lower drag.
Clear-air turbulence experienced during airplane flight, as well as poor astronomical seeing. Most of the terrestrial atmospheric circulation; the oceanic and atmospheric mixed intense oceanic currents. The flow conditions in many industrial equipment and machines; the external flow over all kinds of vehicles such as cars, airplanes and submarines. The motions of matter in stellar atmospheres. A jet exhausting from a nozzle into a quiescent fluid; as the flow emerges into this external fluid, shear layers originating at the lips of the nozzle are created. These layers separate the fast moving jet from the external fluid, at a certain critical Reynolds number they become unstable and break down to turbulence. Biologically generated. Snow fences work by inducing turbulence in the wind, forcing it to drop much of its snow load near the fence. Bridge supports in water. In the late summer and fall, when river flow is slow, water flows smoothly around the support legs. In the spring, when the flow is faster, a higher Reynolds number is associated with the flow.
The flow may start off laminar but is separated from the leg and becomes turbulent. In many geophysical flows, the flow turbulence is dominated by the coherent structures and turbulent events. A turbulent event is a series of turbulent fluctuations that contain more energy than the average flow turbulence; the turbulent events are associated with coherent flow structures such as eddies and turbulent bursting, they play a critical role in terms of sediment scour and transport in rivers as well as contaminant mixing and dispersion in rivers and estuaries, in the atmosphere. In the medical field of cardiology, a stethoscope is used to detect heart sounds and bruits, which are due to turbulent blood flow. In normal individuals, heart sounds are a product of turbulent flow as heart valves close. However, in some conditions turbulent flow can be audible due to other reasons, some of them pathological. For example, in advanced atherosclerosis, bruits can be heard in some vessels that have been narrowed by the disease process.
Turbulence in porous media became a debated subject. Turbulence is characterized by the following features: Irregularity Turbulent flows are always irregular. For this reason, turbulence problems are treated statistically rather than deterministically. Turbulent flow is chaotic. However, not all chaotic flows are turbulent. Diffusivity The available supply of energy in turbulent flows tends to accelerate the homogenization of fluid mixtures; the characteristic, responsible for the enhanced mixing and increased rates of mass and energy transports in a flow is called "diffusivity". Turbulent diffusion is described by a turbulent diffusion coefficient; this turbulent diffusion coefficient is defined in a phenomenological sense, by analogy with the molecular diffusivities, but it does not have a true physical meaning, being dependent on the flow conditions, not a property of the fluid itself. In addition, the turbulent diffusivity concept assumes a con