ARCHIMEDES (c.287 – c.212 BC)

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.

Archimedes’ Principle

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.

Π The Greek symbol pi (enclosed in a picture of an apple) - Pi is a name given to the ratio of the circumference of a circle to the diameterPi

‘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’

Pi-unrolled-720.gif    

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.

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JOHN DEE (1527-1608)

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JOHN DEE

‘Mathematician, cartographer & astronomer. Prolific author, natural magician, alchemist.’

‘Alternative knowledge and methods of learning. ‘Conversations with Angels’. Human power over the world (neo-Platonism).’

Dee was a Hermetic philosopher, a major influence on the ROSICRUCIANS, possibly a spy – astrologer and adviser to Queen Elizabeth I; he chose the day of her coronation.

One of the greatest scholars of his day. His library in his home in Mortlake, London, contained more than 3,000 books.

Greatly influenced by Edward Kelley (1555- 97), whom he met in 1582; from 1583-1589 Dee and Kelley sought the patronage of assorted mid-European noblemen and kings, eventually finding it from the Bohemian Count Vilem Rosenberg.
In 1589, Dee left Kelley to his alchemical research and returned to England where Queen Elizabeth I granted him a position as a college warden; however he had lost respect owing to his occult reputation. Dee returned to Mortlake in 1605 in poor health and increasing poverty and ended his days as a common fortune-teller.

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CHRISTIAN THEOLOGY & WESTERN SCIENCE

bust said to depict a likeness of Socrates

The speculative Greek philosophers, considering the great overarching principles that controlled the Cosmos, were handicapped by a reluctance to test their speculations by experimentation.
At the other end of the spectrum were the craftsmen who fired and glazed pottery, who forged weapons out of bronze and iron. They in turn were hindered by their reluctance to speculate about the principles that governed their craft.

WESTERN SCIENCE is often credited with discoveries and inventions that have been observed in other cultures in earlier centuries.
This can be due to a lack of reliable records, difficulty in discerning fact from legend, problems in pinning down a finding to an individual or group or simple ignorance.

The Romans were technologists and made little contribution to pure science and then from the fall of Rome to the Renaissance science regressed. Through this time, science and technology evolved independently and to a large extent one could have science without technology and technology without science.

Later, there developed a movement to ‘Christianise Platonism’ (Thierry of Chartres).

Platonism at its simplest is the study and debate of the various arguments put forward by the Greek philosopher PLATO (428/7-348/7 BCE).
The philosopher Plotinus is attributed with having founded neo-Platonism, linking Christian and Gnostic beliefs to debate various arguments within their doctrines. One strand of thought linked together three intellectual states of being: the Good (or the One), the Intelligence and the Soul. The neo-Platonic Academy in Greece was closed by the Emperor Justinian in CE 529.
During the early years of the Renaissance, texts on neo-platonism began to be reconsidered, translated and discoursed.

Aristotle’s four causes, from the ‘Timaeus’, were attributed to the Christian God, who works through secondary causes (such as angels).

Efficient Cause – Creator – God the Father

Formal Cause – Secondary agent – God the Son

Material Cause – The four elements: earth, air, fire & water.
Because these four are only fundamental forms of the single type of matter, they cannot be related to any idea of ‘elements’ as understood by modern science – they could be transmuted into each other. Different substances, although composed of matter would have different properties due to the differing amounts of the qualities of form and spirit. Thus a lump of lead is made of the same type of matter (fundamental form) as a lump of gold, but has a different aggregation of constituents. Neither lead nor gold would contain much spirit – not as much as air, say, and certainly not as much as God, who is purely spiritual. ( ALCHEMY )

Final Cause – Holy Spirit

All other is ‘natural’ – underwritten by God in maintaining the laws of nature without recourse to the supernatural.
Science was the method for investigating the world. It involved carrying out careful experiments, with nature as the ultimate arbiter of which theories were right and which were wrong.

Robert Grosseteste (1168-1253) Bishop of Lincoln (Robert ‘Bighead’) – neo-Platonic reading of Genesis – emanation of God’s goodness, like light, begins creation. Light is thus a vehicle of creation and likewise knowledge (hence ‘illumination’), a dimensionless point of matter with a dimensionless point of light superimposed upon it (dimensions are created by God). Spherical radiation of light carries matter with it until it is dissipated. Led to studies of optical phenomena (rainbow, refraction, reflection).

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ROBERT GROSSETESTE

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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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THOMAS HUNT-MORGAN (1866-1945)

1933 – USA

‘‘The Mechanism of Mendelian Heredity’ (1915), ‘The Theory of the Gene’ (1926)’

Morgan laid the foundation for understanding MENDEL’s observations and helped to provide the science required to reinforce CHARLES DARWIN’s conclusions.

Starting with Mendel’s laws of segregation and independent assortment, Morgan investigated why there are far fewer chromosomes – the long thread-like structures present in the nucleus of every living cell, which grow and divide during cell splitting, – than there are ‘units of heredity’. Morgan could not see how these few chromosomes could account for all the changes that occur from one generation to the next.

Mendel’s ‘factors of heredity’ had been renamed ‘genes’ in 1909 by the Dane Wilhelm Johannsen.

When the organism forms its reproductive cells (gametes), the genes segregate and pass to different gametes.
Since it had been separately established that chromosomes play an important part in inheritance, then groups of genes had to be present on a single chromosome.
If all the genes were arranged along chromosomes, and all chromosomes were transmitted intact from one generation to the next, then many characteristics would be inherited together. This implicitly invalidates Mendel’s law of independent assortment, which dictated that hereditary traits caused by genes would occur in all possible mathematical combinations in a series of descendants, independent of each other.

Experimental evidence often seemed to back-up the mathematical forecasts for characteristics present in descendants that Mendel had suggested; Morgan felt that the law of independent assortment could not accurately model the process of arriving at the end result.

He began his experiments with the fruit fly, which has just four pairs of chromosomes, in 1908.
He observed a mutant white-eyed male fly, which he extracted for breeding with ordinary red-eyed females. Over subsequent generations of interbred offspring, the white-eyed trait returned in some descendants, all of which turned out to be males. Clearly, certain genetic traits were not occurring independently of each other but were being passed on in groups.
Rather than invalidating Mendel’s law of independent assortment, a simple adjustment was required to unite it with Hunt’s belief in chromosomes to produce his thesis.
He suggested that the law of independent assortment did apply – but only to genes found on different chromosomes. For those on the same chromosome, linked traits would be passed on; usually a sex-related factor with other specific features (such as, the male sex and the white-eyed characteristic).

The results of his work convinced Morgan that genes were arranged on chromosomes in a linear manner and could be mapped. Further testing showed that, as chromosomes actually break apart and re-form during the production of sperm and egg cells, linked traits could occasionally be broken during the exchange of genes (recombination) that occurred between pairs of chromosomes during the process of cell division. He hypothesised that the nearer on the chromosome the genes were located to each other, the less likely the linkages were to be broken. Thus by measuring the occurrence of breakages he could work out the position of the genes along the chromosome.
In 1911 he produced the first chromosome map showing the position of five genes linked to gender characteristics.

In 1933 Hunt Morgan received the Nobel Prize for Physiology.

picture of the Nobel medal - link to nobelprize.org

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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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False positives: fraud and misconduct are threatening scientific research | Science | The Guardian

False positives: fraud and misconduct are threatening scientific research | Science | The Guardian.

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