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    LEONHARD EULER (1707- 83)

    1755 – Switzerland

    ‘Analytical calculus – the study of infinite processes and their limits’

    Swiss mathematician. His notation is even more far-reaching than that of LEIBNIZ and much of the mathematical notation that is in use to-day may be credited to Euler.

    The number of theorems, equations and formulae named after him is enormous.
    Euler made important discoveries in the analytic geometry of surfaces and the theory of differential equations.

    Euler popularised the use of the symbol ‘Π‘ (Pi); e  , for the base of the natural logarithm; and i , for the imaginary unit.
    Euler is credited with contributing the useful notations  f (x) , for the general function of x ; and  Σ , to indicate a general sum of terms.

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    • Leonhard Euler (usna.edu/)
    • Quote (boyslumber.wordpress.com)
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    Mathematicians

    picture of mathematician Pythagoras

    picture of mathematician Euclid

    picture of mathematician Archimedes

    picture of mathematician Hipparchus

    picture of mathematician Al Khwarizmi

    Image of the head of statue of Fibonacci


    picture of mathematician Descartes

    image depicting Pierre de Fermat ©

    picture of mathematician Pascal

    picture of mathematician Carl Gauss

    picture of mathematician Gottfried Liebniz

    image of David Hilbert made from the Hilbert curve

    portrait of Paul Dirac © neon designs 2012

    picture of mathematician Isaac Newton

    picture of mathematician Alan Turing

    Statue of Janus Bolyai©

    picture of mathematician Daniel Bernoulli

    picture of mathematician George Boole

    picture of mathematician Ludwig Boltzman

    Bust depicting Evariste Galois©

    Picture of Osborne Reynolds&copy

    Portrait of Leonhard Euler (1707-83)©

    picture of mathematician Kurt Godel

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    HIPPARCHUS (c.190 – c.125 BC)

    134 BCE – Nicea, Turkey

    ‘Observation of a new star in the constellation Scorpio’

    The ‘Precession of the Equinoxes’

    By the time Hipparchus was born, astronomy was already an ancient art.

    Hipparchus plotted a catalogue of the stars – despite warnings that he was thus guilty of impiety. Comparing his observations with earlier recordings from Babylonia he noted that the celestial pole changed over time.
    He speculated that the stars are not fixed as had previously been thought and recorded the positions of 850 stars.

    Hipparchus‘ astronomical calculations enabled him to plot the ecliptic, which is the path of the Sun through the sky. The ecliptic is at an angle to the Earth‘s equator, and crosses it at two points, the equinoxes (the astronomical event when the Sun is at zenith over the equator, marking the two occasions during the year when both hemispheres are at right angles to the Sun and day and night are of equal length).

    The extreme positions of summer and winter mark the times in the Earth’s orbit where one of the hemispheres is directed towards or away from the Sun.

    Solstice

    The Sun is furthest away at the solstices.

    From his observations, he was able to make calculations on the length of the year.
    There are several ways of measuring a year astronomically and Hipparchus measured the ‘tropical year’, the time between equinoxes.

    Hipparchus puzzled that even though the Sun apparently traveled a circular path, the seasons – the time between the solstices and equinoxes – were not of equal length. Intrigued, he worked out a method of calculating the Sun’s path that would show its exact location on any date.

    To facilitate his celestial observations he developed an early version of trigonometry.
    With no notion of sine, he developed a table of chords which calculated the relationship between the length of a line joining two points on a circle and the corresponding angle at the centre.

    By comparing his observations with those noted by Timocharis of Alexandria a century and a half previously, Hipparchus noted that the points at which the equinox occurred seemed to move slowly but consistently from east to west against the backdrop of fixed stars.

    We now know that this phenomenon is not caused by a shift in the stars.
    Because of gravitational effects, over time the axis through the geographic North and South poles of the Earth points towards different parts of space and of the night sky.
    The Earth’s rotation experiences movement caused by a slow change in the direction of the planet’s tilt; the axis of the Earth ‘wobbles’, or traces out a cone, changing the Earth’s orientation as it orbits the Sun.
    The shift in the orbital position of the equinoxes relative to the Sun and the change in the seasons is now known as ‘the precession of the equinoxes’, but Hipparchus was basically right.

    Hipparchus‘ only large error was to assume, like all those of his time except ARISTARCHUS that the Earth is stationary and that the Sun, moon, planets and stars revolve around it. The fact that the stars are fixed and the Earth is moving makes such a tiny difference to the way the Sun, moon and stars appear to move that Hipparchus was still able to make highly accurate calculations.

    These explanations may show how many people become confused by claims that the Earth remains stationary as was believed by the ancients – from our point-of-view on Earth that IS how things could appear.
    a) demonstration of precession.

    youtube=https://www.youtube.com/watch?v=qlVgEoZDjok
    b) demonstration of the equinoxes, but not of the precession, which takes place slowly over a cycle of 26,000 years.

    youtube=http://www.youtube.com/watch?v=q4_-R1vnJyw&w=420&h=315

    Because the Babylonians kept records dating back millennia, the Greeks were able to formulate their ideas of the truth.

    Hipparchus gave a value for the annual precession of around 46 seconds of arc (compared to a modern figure of 50.26 seconds). He concluded that the whole star pattern was moving slowly eastwards and that it would revolve once every 26,000 years.

    Hipparchus also made observations and calculations to determine the orbit of the moon, the dates of eclipses and devised the scale of magnitude or brightness that, considerably amended, is still in use.

    PTOLEMY cited Hipparchus as his most important predecessor.

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    GREGOR MENDEL (1822- 84)

    1865 – Austria

    • ‘Law of Segregation: In sexually reproducing organisms, two units of heredity control each trait. Only one of such units can be represented in a single sexually reproductive cell’

    • ‘Law of Independent Assortment: Each of a pair of contrasted traits may be combined with either of another pair’

    These laws laid the foundation for the science of genetics.

    The biologist Lamarck (1744-1829) had proposed a theory of inheritance of acquired characteristics and had suggested that inherited characteristics are influenced by environment. Mendel planted an atypical variety of an oriental plant next to a typical variety – the offspring retained the essential traits of their parents, which meant that the characteristics that were inherited were not influenced by the environment. This simple test led Mendel to embark on the path that would lead to the discovery of the laws of heredity.

    Mendel’s aim was to discover ” a generally applicable law of the formation and development of hybrids “. He addressed this by studying the effect of cross-breeding on seven pairs of contrasting characteristics of Pisum sativum, a strain of pea.
    His work on peas indicated that features of the plant; seed shape, seed colour, pod shape, pod colour, flower colour, flower position and stem length; were passed on from one generation to the next by some physical element. He realised that each characteristic of a plant was inherited independently, and that the ratios of plants exhibiting each trait could be statistically predicted.

    photograph of GREGOR MENDEL ©

    GREGOR MENDEL

    A common assumption in Mendel’s time was that when two alternative features were combined, an average of these features would occur. For example, a tall plant and a short one would result in medium height offspring. For seven years Mendel kept an exact record of the inherited characteristics of 28,000 pea plants, taking great pains to avoid accidental cross-fertilization; then he applied mathematics to the results. These quantitative data allowed him to see statistical patterns and ratios that had eluded his predecessors.

    From his analysis he found that certain characteristics of plants are due to factors passed intact from generation to generation.
    Mendel observed that individual plants of the first generation of hybrids (crossbred plants) usually showed the traits of only one parent. The crossing of yellow seeded plants with green seeded ones gave rise to yellow seeds; the crossing of tall stemmed ones with short-stemmed varieties gave rise to tall-stemmed plants.

    The factors determining a trait are passed on to the offspring during reproduction.

    Mendel worked out that the factors for each trait are grouped together in pairs and that the offspring receives one part of a pair from each parent.

    Contrary to the popular belief of the time, these factors do not merge. Any individual pea always exhibits one trait or the other, never a mixture of the two possible expressions of the trait; only one trait from each pair of factors donated by the parents would be expressed in the offspring, although there are four possible combinations of factors.
    This is now described as Mendel’s law of segregation.
    An offspring inherits from its parents either one trait or the other, but not both.

    He decided that some factors were ‘dominant’ and some were ‘recessive’ and was able to conclude that certain expressed traits, such as yellow seeds or tall stems, were the dominant ones and that other traits, such as shortness of stem and green seeds, were recessive. It appeared that the dominant factors consumed or destroyed the recessive factors – but this could not be the case, as the second generation of hybrids exhibited both the dominant and recessive traits of their ‘grandparents’. Across a series of generations of descendants, plants did not average out to a medium, but instead inherited the original features (for example, either tallness or shortness) in consistent proportions, a ratio of 3:1, according to the dominant factor.
    The 3:1 ratio would apply because the dominant factor would feature whenever it was present.

    He also noted that the different pairs of factors making up the characteristics of the pea plant ( such as the pair causing flower colour, the pair causing seed shape and so on ), when crossed, occurred in all possible mathematical combinations. This convinced him that the elements regulating the different features acted independently of each other, so the inheritance of one particular colour of flower was not influenced, for example, by the inheritance of pea shape.
    This is now described as Mendel’s law of independent assortment.

    He first articulated his results in 1865 and in 1866, which was shortly after Darwin’s ‘Origin of Species’ appeared, published them in an article ‘Versuche über Pflanzen-Hybriden’ (Experiments with plant hybrids).

    No one before him had attempted to use mathematics and statistics as a means of understanding and predicting biological processes and during his lifetime and for some time after, his results were largely ignored.

    Around the time of Mendel’s death, scientists using ever improving optics to study the minute architecture of cells coined the term ‘chromosome’ to describe the long, stringy bodies in the cell nucleus.

    The seven traits studied in peas
    TRAIT DOMINANT TRAIT RECESSIVE TRAIT
    Type of seed surface smooth wrinkled
    Colour of seed albumen yellow green
    Colour of seed coat grey white
    Form of ripe pod inflated constricted
    Colour of unripe pod green yellow
    Position of flowers on stem axial terminal
    Length of stem tall short

    ‘There is doubt as to the probity of this Jesuit scholar, some claiming that his data was falsified whilst others argue that it is accurate’
    Pilgrim, I. (1984) The Too-Good-to-be-True Paradox and Gregor Mendel. Journal of Heredity,#75, pp 501-2. Cited in Brake,M.L. & Hook, N. Different Engines – How science drives fiction and fiction drives science

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    ALBERT EINSTEIN (1879-1955)

    1905 – Switzerland

    1. ‘the relativity principle: All laws of science are the same in all frames of reference.
    2. constancy of the speed of light: The speed of light in a vacuüm is constant and is independent of the speed of the observer’
    photo portrait of Albert Einstein &copy:

    EINSTEIN

    The laws of physics are identical to different spectators, regardless of their position, as long as they are moving at a constant speed in relation to each other. Above all the speed of light is constant. Classical laws of mechanics seem to be obeyed in our normal lives because the speeds involved are insignificant.

    Newton’s recipe for measuring the speed of a body moving through space involved simply timing it as it passed between two fixed points. This is based on the assumptions that time is flowing at the same rate for everyone – that there is such a thing as ‘absolute’ time, and that two observers would always agree on the distance between any two points in space.
    The implications of this principle if the observers are moving at different speeds are bizarre and normal indicators of velocity such as distance and time become warped. Absolute space and time do not exist. The faster an object is moving the slower time moves. Objects appear to become shorter in the direction of travel. Mass increases as the speed of an object increases. Ultimately nothing may move faster than or equal to the speed of light because at that point it would have infinite mass, no length and time would stand still.

    ‘The energy (E) of a body equals its mass (m) times the speed of light (c) squared’

    This equation shows that mass and energy are mutually convertible under certain conditions.

    The mass-energy equation is a consequence of Einstein’s theory of special relativity and declares that only a small amount of atomic mass could unleash huge amounts of energy.

    Two of his early papers described Brownian motion and the ‘photoelectric’ effect (employing PLANCK’s quantum theory and helping to confirm Planck’s ideas in the process).

    1915 – Germany

    ‘Objects do not attract each other by exerting pull, but the presence of matter in space causes space to curve in such a manner that a gravitational field is set up. Gravity is the property of space itself’

    From 1907 to 1915 Einstein developed his special theory into a general theory that included equating accelerating forces and gravitational forces. This implies light rays would be bent by gravitational attraction and electromagnetic radiation wavelengths would be increased under gravity. Moreover, mass and the resultant gravity, warps space and time, which would otherwise be ‘flat’, into curved paths that other masses (e.g. the moons of planets) caught within the field of the distortion follow. The predictions from special and general relativity were gradually proven by experimental evidence.

    Einstein spent much of the rest of his life trying to devise a unified theory of electromagnetic, gravitational and nuclear fields.

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    WERNER HEISENBERG (1901- 76)

    1927 – Germany

    ‘It is impossible to determine exactly both the position and momentum of a particle (such as an electron) simultaneously’

    The principle excludes the existence of a particle that is stationary.

    To measure both the position and momentum ( momentum = mass × velocity ) of a particle simultaneously requires two measurements: the act of performing the first measurement will disturb a particle and so create uncertainty in the second measurement.
    Thus the more accurately a position is known; the less accurately can the momentum be determined.

    The disturbance is so small it can be ignored in the macroscopic world, but is quite dramatic for particles in the microscopic world.
    MAX BORN’S ‘probabilistic’ interpretation, expressed at about the same time, concerning the likelihood of finding a particle at any point through probability defined by the amplitude of its associated wave, led to similar conclusions.

    The uncertainty principle also applies to energy and time. A particle’s kinetic energy cannot be measured with complete precision either.

    Heisenberg suggested the model of the proton and neutron being held together in the nucleus of the atom after the work of JAMES CHADWICK who discovered the neutron in 1932.

    Heisenberg decided to try to develop a new model of the atom, more fundamentally based on quantum theory that worked for all atoms. He believed the approach of trying to visualise a physical model of the atom was destined to fail because of the paradoxical wave-particle nature of electrons.

    Every particle has an associated wave. The position of a particle can be precisely located where the wave’s undulations are most intense. But where the wave’s undulations are most intense, the wavelength is also at its most ill-defined, and the velocity of the associated particle is impossible to determine. Similarly, a particle with a well-defined wavelength has a precise velocity but a very ill-defined position.

    Since the orbits of electrons could not be observed, he decided to ignore them and focus instead on what could be observed and measured; namely the energy they emitted and absorbed, as shown in the spectral lines. He tried to devise a mathematical way of representing the orbits of electrons, and to use this as a way of predicting the atomic features shown up in the spectral lines.
    He showed that matrix mechanics could account for many of the properties of atoms, including those with more than one electron.

    Together with PAUL DIRAC, Pascual Jordan created a new set of equations based on the rival theories of Schrödinger and Heisenberg, which they called ‘transformation theory’. Whilst studying these equations, Heisenberg noticed the paradox that measurements of position and velocity (speed and direction) of particles taken at the same time gave imprecise results. He believed that this uncertainty was a part of the nature of the sub-atomic world. The act of measuring the velocity of a subatomic particle will change it, making the simultaneous measurement of its position invalid.

    An unobserved object is both a particle and a wave. If an experimenter chooses to measure the object’s velocity, the object will transform itself into a wave. If an experimenter chooses to measure its position, it will become a particle. By choosing to observe either one thing or the other, the observer is actually affecting the form the object takes.
    The practical implication of this is that one can never predict where an electron will be at a precise moment, one can only predict the probability of its being there.

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