Third Century BCE – Syracuse (a Greek city in Sicily)
‘Archimedes’ Screw – a device used to pump water out of ships and to irrigate fields’
Archimedes investigated the principles of static mechanics and pycnometry (the measurement of the volume or density of an object). He was responsible for the science of hydrostatics, the study of the displacement of bodies in water.
Buoyancy – ‘A body fully or partially immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced by the body’
The upthrust (upward force) on a floating object such as a ship is the same as the weight of water it displaces. The volume of the displaced liquid is the same as the volume of the immersed object. This is why an object will float. When an object is immersed in water, its weight pulls it down, but the water, as Archimedes realised, pushes back up with a force that is equal to the weight of water the object pushes out-of-the-way. The object sinks until its weight is equal to the upthrust of the water, at which point it floats.
Objects that weigh less than the water displaced will float and objects that weigh more will sink. Archimedes showed this to be a precise and easily calculated mathematical principle.
Syracuse’s King Hiero, suspecting that the goldsmith had not made his crown of pure gold as instructed, asked Archimedes to find out the truth without damaging the crown.
Archimedes first immersed in water a piece of gold that weighed the same as the crown and pointed out the subsequent rise in water level. He then immersed the crown and showed that the water level was higher than before. This meant that the crown must have a greater volume than the gold, even though it was the same weight. Therefore it could not be pure gold and Archimedes thus concluded that the goldsmith had substituted some gold with a metal of lesser density such as silver. The fraudulent goldsmith was executed.
Archimedes came to understand and explain the principles behind the compound pulley, windlass, wedge and screw, as well as finding ways to determine the centre of gravity of objects.
He showed that the ratio of weights to one another on each end of a balance goes down in exact mathematical proportion to the distance from the pivot of the balance.
Perhaps the most important inventions to his peers were the devices created during the Roman siege of Syracuse in the second Punic war.
He was killed by a Roman soldier during the sack of the city.
‘All circles are similar and the ratio of the circumference to the diameter of a circle is always the same number, known as the constant, Pi’
The Greek tradition disdained the practical. Following PLATO the Greeks believed pure mathematics was the key to the perfect truth that lay behind the imperfect real world, so that anything that could not be completely worked out with a ruler and compass and elegant calculations was not true.
In the eighteenth century AD the Swiss mathematician LEONHARD EULER was the first person to use the letter Π , the initial letter of the Greek word for perimeter, to represent this ratio.
The earliest reference to the ratio of the circumference of a circle to the diameter is an Egyptian papyrus written in 1650 BCE, but Archimedes first calculated the most accurate value.
He calculated Pi to be 22/7, a figure which was widely used for the next 1500 years. His value lies between 3 1/2 and 3 10/71, or between 3.142 and 3.141 accurate to two decimal places.
‘The Method of Exhaustion – an integral-like limiting process used to compute the area and volume of two-dimensional lamina and three-dimensional solids’
Archimedes realised how much could be achieved through practical approximations, or, as the Greeks called them, mechanics. He was able to calculate the approximate area of a circle by first working out the area of the biggest hexagon that would fit inside it and then the area of the smallest that would fit around it, with the idea in mind that the area of the circle must lie approximately halfway between.
By going from hexagons to polygons with 96 sides, he could narrow the margin for error considerably. In the same way he worked out the approximate area contained by all kinds of different curves from the area of rectangles fitted into the curve. The smaller and more numerous the rectangles, the closer to the right figure the approximation became.
This is the basis of what thousands of years later came to be called integral calculus.
Archimedes’ reckonings were later used by Kepler, Fermat, Leibniz and Newton.
In his treatise ‘On the Sphere and the Cylinder’, Archimedes was the first to deduce that the volume of a sphere is 4/3 Pi r3 where r is the radius.
He also deduced that a sphere’s surface area can be worked out by multiplying that of its greatest circle by four; or, similarly, a sphere’s volume is two-thirds that of its circumscribing cylinder.
Like the square and cube roots of 2, Pi is an irrational number; it takes a never-ending string of digits to express Pi as a number.
It is impossible to find the exact value of Pi – however, the value can be calculated to any required degree of accuracy.
In 2002 Yasumasa Kanada (b.1949) of Tokyo University used a supercomputer with a memory of 1024GB to compute the value to 124,100,000,000 decimal places. It took 602 hours to perform the calculation.
- Pi (math.com)
1931 – USA
‘A framework for understanding the electronic and geometric structure of molecules and crystals’
An important aspect of this framework is the concept of hybridisation: in order to create stronger bonds, atoms change the shape of their orbitals (the space around a nucleus in which an electron is most likely to be found) into petal shapes, which allow more effective overlapping of orbitals.
A chemical bond is a strong force of attraction linking atoms in a molecule or crystal. BOHR had already shown that electrons inhabit fixed orbits around the nucleus of the atom. Atoms strive to have a full outer shell (allowed orbit), which gives a stable structure. They may share, give away or receive extra electrons to achieve stability. The way atoms will form bonds with others, and the ease with which they will do it, is determined by the configuration of electrons.
Earlier in the century, Gilbert Lewis (1875-1946) had offered many of the basic explanations for the structural bonding between elements, including the sharing of a pair of electrons between atoms and the tendency of elements to combine with others to fill their electron shells according to rigidly defined orbits (with two electrons in the closest orbit to the nucleus, eight in the second orbit, eight in the third and so on).
Pauling was the first to enunciate an understanding of a physical interpretation of the bonds between molecules from a chemical perspective, and of the nature of crystals.
In a covalent bond, one or more electrons are shared between two atoms. So two hydrogen atoms form the hydrogen molecule, H2, by each sharing their single electron. The two atoms are bound together by the shared electrons. This was proposed by Lewis and Irving Langmuir in 1916.
In an ionic bond, one atom gives away one or more electrons to another atom. So in common salt, sodium chloride, sodium gives away its spare electron to chlorine. As the electron is not shared, the sodium and chlorine atoms are not bound together in a molecule. However, by losing an electron, sodium acquires a positive charge and chlorine, by gaining an electron, acquires a negative charge. The resulting sodium and chlorine ions are held in a crystalline structure. Until Pauling’s explanation it was thought that they were held in place only by electrical charges, the negative and positive ions being drawn to each other.
Pauling’s work provided a value for the energy involved in the small, weak hydrogen bond.
When a hydrogen atom forms a bond with an atom which strongly attracts its single electron, little negative charge is left on the opposite side of the hydrogen atom. As there are no other electrons orbiting the hydrogen nucleus, the other side of the atom has a noticeable positive charge – from the proton in the nucleus. This attracts nearby atoms with a negative charge. The attraction – the hydrogen bond – is about a tenth of the strength of a covalent bond.
In water, attraction between the hydrogen atoms in one water molecule and the oxygen atoms in other water molecules makes water molecules ‘sticky’. It gives ice a regular crystalline structure it would not have otherwise. It makes water liquid at room temperature, when other compounds with similarly small molecules are gases at room temperature.
One aspect of the revolution he brought to chemistry was to insist on considering structures in terms of their three-dimensional space. Pauling showed that the shape of a protein is a long chain twisted into a helix or spiral. The structure is held in shape by hydrogen bonds.
He also explained the beta-sheet, a pleated sheet arrangement given strength by a line of hydrogen bonds.
He devised the electronegativity scale, which ranks elements in order of their electronegativity – a measure of the attraction an atom has for the electrons involved in bonding (0.7 for caesium and francium to 4.0 for fluorine). The electronegativity scale lets us say how covalent or ionic a bond is.
Pauling’s application of quantum theory to structural chemistry helped to establish the subject. He took from quantum mechanics the idea of an electron having both wave-like and particle-like properties and applied it to hydrogen bonds. Instead of there being just an electrical attraction between water molecules, Pauling suggested that wave properties of the particles involved in hydrogen bonding and those involved in covalent bonding overlap. This gives the hydrogen bonds some properties of covalent bonds.
1922 – while investigating why atoms in metals arrange themselves into regular patterns, Pauling used X-ray diffraction at CalTech to determine the structure of molybdenum.
When X-rays are directed at a crystal, some are knocked off course by striking atoms, while others pass straight through as if there are no atoms in their path. The result is a diffraction pattern – a pattern of dark and light lines that reveal the positions of the atoms in the crystal.
Pauling used X-ray and electron diffraction, magnetic effects and measurements of the heat of chemical reactions to calculate the distances and angles between atoms forming bonds. In 1928 he published his findings as a set of rules for working out probable crystalline structures from the X-ray diffraction patterns.
1939 – ‘The Nature of the Chemical Bond and the Structure of Molecules’
Pauling suggests that in order to create stronger bonds, atoms change the shapes of their waves into petal shapes; this was the ‘hydridisation of orbitals’.
Describing hybridisation, he showed that the labels ‘ionic’ and ‘covalent’ are little more than a convenience to group bonds that really lie on a continuous spectrum from wholly ionic to wholly co-valent.
Pauling developed six key rules to explain and predict chemical structure. Three of them are mathematical rules relating to the way electrons behave within bonds, and three relate to the orientation of the orbitals in which the electrons move and the relative position of the atomic nuclei.
As carbon has four filled and four unfilled electron shells it can form bonds in many different ways, making possible the myriad organic compounds found in plants and animals. The concept of hybridisation proved useful in explaining the way carbon bonds often fall between recognised states, which opened the door to the realm of organic chemistry.
X-ray diffraction alone is not very useful for determining the structure of complex organic molecules, but it can show the general shape of the molecule. Pauling’s work showed that physical chemistry at the molecular level could be used to solve problems in biology and medicine.
A problem that needed resolving was the distance between particular atoms when they joined together. Carbon has four bonds, for instance, while oxygen can form two.It would seem that in a molecule of carbon dioxide, which is made of one carbon and two oxygen atoms, two of carbon’s bonds will be devoted to each oxygen.
Well-established calculations gave the distance between the carbon and oxygen atoms as 1.22 × 10-10m. Analysis gave the size of the bond as 1.16 Angstroms. The bond is stronger, and hence shorter. Pauling’s quantum .3-2. explanation was that the bonds within carbon dioxide are constantly resonating between two alternatives. In one position, carbon makes three bonds with one of the oxygen molecules and has only one bond with the other, and then the situation is reversed.
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1850 – France
‘A Foucault pendulum is a simple pendulum – a long wire with a heavy weight (bob) at the end – except that at the top it is attached to a joint which allows it to swing in any direction’
Foucault’s pendulum proved that the Earth is rotating
Once a Foucault pendulum is set in motion, it seems not to swing back and forth in the same direction but to rotate. In fact, it is the rotation of the Earth beneath the pendulum which gives rise to its apparent rotation.
The angle of rotation per hour, which is constant at any particular location, can be calculated from the formula 15 sin Φ, where Φ is the geographical latitude of the observer. At the North or South Pole, the pendulum would rotate through 360 degrees once a day. At the equator it would not rotate at all.
1995 – England
‘A slice of toast sliding off a plate or table has a natural tendency to land butter side down’
This provides prima facie evidence for Murphy’s law. Matthews writes in a detailed research paper ‘Tumbling Toast, Murphy’s Law and the Fundamental Constants‘ in the European Journal of Physics (July 1995) ‘Toast does indeed have a natural tendency to land butter side down, essentially because the gravitational torque induced as the toast topples over the edge of the plate/table is insufficient to bring the toast butter-side up again by the time that it hits the floor’. The argument was explained by five pages of mathematical calculations. Matthew’s extraordinary insights into the behaviour of buttered toast won him the 1996 Ig Nobel Prize for physics.
In 2001 Matthews tried to prove his theory experimentally. About 1000 schoolchildren from schools across the UK took part in his experiments and performed 9821 drops of toast, of which 6101 were butter-side-down landings – ‘And thus Robert Matthews demonstrated both theoretically and experimentally that nature abhors a newly vacuumed floor’.
‘A given amount of work produces a specific amount of heat’
4.18 joules of work is equivalent to one calorie of heat.
In 1798 COUNT RUMFORD suggested that mechanical work could be converted into heat. This idea was pursued by Joule who conducted thousands of experiments to determine how much heat could be obtained from a given amount of work.
Even in the nineteenth century, scientists did not fully understand the properties of heat. The common belief held that it was some form of transient fluid – retained and released by matter – called CALORIC. Gradually, the idea that it was another form of energy, expressed as the movement of molecules gained ground.
Heat is now regarded as a mode of transfer of energy – the transfer of energy by virtue of a temperature difference. Heat is the name of a process, not that of an entity.
Joule began his experiments by examining the relationship between electric current and resistance in the wire through which it passed, in terms of the amount of heat given off. This led to the formulation of Joule’s ideas in the 1840s, which mathematically determined the link.
Joule is remembered for his description of the conversion of electrical energy into heat; which states that the heat (Q) produced when an electric current (I) flows through a resistance (R) for a time (t) is given by Q=I2Rt
Its importance was that it undermined the concept of ‘caloric’ as it effectively determined that one form of energy was transforming itself into another – electrical energy to heat energy. Joule proved that heat could be produced from many different types of energy, including mechanical energy.
Joule was the son of a brewer and all his experiments on the mechanical equivalent of heat depended upon his ability to measure extremely slight increases in temperature, using the sensitive thermometers available to him at the brewery. He formulated a value for the work required to produce a unit of heat. Performing an improved version of Count Rumford’s experiment, he used weights on a pulley to turn a paddle wheel immersed in water. The friction between the water and the paddle wheel caused the temperature of the water to rise slightly. The amount of work could be measured from the weights and the distance they fell, the heat produced could be measured by the rise in temperature.
Joule went on to study the role of heat and movement in gases and subsequently with WILLIAM THOMSON, who later became Lord Kelvin, described what became known as the ‘Joule-Thomson effect’ (1852-9). This demonstrated how most gases lose temperature on expansion due to work being done in pulling the molecules apart.
Thomson thought, as CARNOT had, that heat IN equals heat OUT during a steam engine’s cycle. Joule convinced him he was wrong.
The essential correctness of Carnot’s insight is that the work performed in a cycle divided by heat input depends only on the temperature of the source and that of the sink.
Synthesising Joule’s results with Carnot’s ideas, it became clear that a generic steam engine’s efficiency – work output divided by heat input – differed from one (100%) by an amount that could be expressed either as heat OUT at the sink divided by heat IN at the source, or alternatively as temperature of the sink divided by temperature of the source. Carnot’s insight that the efficiency of the engine depends on the temperature difference was correct. Temperature has to be measured using the right scale. The correct one had been hinted at by DALTON and GAY-LUSSAC’s experiments, in which true zero was minus 273degrees Celsius.
A perfect cyclical heat engine with a source at 100degrees Celsius and a sink at 7degrees has an efficiency of 1 – 280/373. The only way for the efficiency to equal 100% – for the machine to be a perfect transformer of heat into mechanical energy – is for the sink to be at absolute zero temperature.
Joule’s work helped in determining the first law of thermodynamics; the principle of the conservation of energy. This was a natural extension of his work on the ability of energy to transform from one type to another.
Joule contended that the natural world has a fixed amount of energy which is never added to nor destroyed, but which just changes form.
The SI unit of work and energy is named the joule (J).
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