‘At noon on the day of the summer solstice, the Sun is directly overhead in Syene (now Aswan) and there is no shadow, but at the same time in Alexandria the Sun is at an angle and there is a measurable shadow’

Eratosthenes used this concept to calculate the circumference of the Earth.

In 230 BCE, the Greek philosopher Eratosthenes worked out the circumference of the Earth to be 25,000 miles (40,000 km) by studying shadows cast by the Sun in both Alexandria and Syene on the day of the summer solstice. Eratosthenes knew from his predecessors that at noon on the longest day of the year (the summer solstice), the Sun would be directly overhead at Syene when a vertical post would cast no shadow, whereas a post in Alexandria 800 kilometers to the north would have a measurable shadow

Eratosthenes reasoned that the surface of the Earth was curved, resulting in the Sun’s rays being different in different locations. With the aid of simple geometrical instruments he found that in Alexandria at noon the Sun’s rays were falling at an angle of 7.2 degrees, which is one fiftieth of 360 degrees. Having determined the difference in the angles between the axes of the two posts, these axes, if extrapolated downwards would meet at the centre of a spherical Earth. Knowing the distance between the two places, he calculated that the circumference of the Earth was fifty times that distance.

As 7 degrees is approximately one-fiftieth of a circle, multiplying the 800 km distance between the posts by 50 gives a circumference for the Earth of 40,000 km and dividing by pi gives a diameter of 12,800 km.

Eratosthenes’ value comes to 39,350 kilometres, compared to a true average length of 40,033 kilometres.

Eratosthenes was a scholar, an astronomer, mathematician, geographer, historian, literary critic and poet. He was nicknamed ‘Beta’ (the second letter of the Greek alphabet) because he was considered the second best at everything.

“Epicurus’s philosophy combines a physics based on an atomistic materialism with a rational hedonistic ethics that emphasizes moderation of desires and cultivation of friendships.”

Summarized by the Roman author Lucretius, who wrote ‘On the Nature of the Universe’ in 55 BCE – “The light and heat of the Sun; these are composed of minute atoms which, when they are shoved off, lose no time in shooting right across the interspace of air in the direction imparted by the shove”. This may be considered as accurate for the time, when most people thought that sight was associated with something reaching out from the eye (EMPEDOCLES) .

Plato wrote of a marriage between the inner light and the outer light.

Euclid worried about the speed with which sight worked. He pointed out that if you close your eyes, then open them again, even the distant stars reappear immediately in your sight, although the influence of sight has had to travel all the way from your eyes to the stars and back again before you could see them.

‘Observation of a new star in the constellation Scorpio’

The ‘Precession of the Equinoxes’

HIPPARCHUS

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.

Schematic presentation of a seasonal cycle

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 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.

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.

– throughout the Middle Ages, astrology and astronomy were closely linked in both the Western and the Arabic worlds.
Although astrology was used for prediction, pre-modern astrology required a substantial command of mathematics and an informed astronomical knowledge.

PTOLEMY – ‘ The Almagest ’ how the planets move; ‘ Tetrabiblos ’ what effect the qualities of the planets (Mars – hot & dry, Moon – cold & wet [affect on the tides]) and their relative positions will have.

Belief that the influence of the planets may have an effect on earthly health and other matters (disease and character traits).

Tables of positions of planets became developed from the Babylonian originals in the Islāmic world.

Alphonsine tables produced for King Alphonso X of Castile in 1275.

Prognostication repeatedly condemned by the Church as influence of the planets denies the concept of free will.

Refutation of astrology is difficult owing to its complexity.

A great deal of what we know about the ancient world and its scientific ideas has come to us from documents which were translated from ancient Greek or other ancient languages into Arabic, and later from Arabic into European languages. The material reached the Arab world in many cases through the Roman empire in the East, Byzantium, which survived until 1453, almost a thousand years after the fall of Rome, during the period known in Europe as the Dark Ages.
During this time the consolidating influence of Islāmic religion saw Arab Muslims begin to build an empire that was to stretch across the Middle East and across North Africa into Spain. At the heart of the Islāmic world the caliphs ruled in Baghdad. Arab scientists sowed the seeds that would later be reaped in the scientific revolution of the seventeenth century, especially under the Abbasid dynasty during the caliphate of Harun al-Rashid and his son al-Mamun, and the Middle East became the intellectual hub of the World.

In the ninth century, at the House of Wisdom – a mixture of library, research institute and university – scholars worked to translate the great works of the GREEK thinkers. Muslim scholars of this golden age made important and original contributions to mathematics and astronomy, medicine and chemistry. They developed the ASTROLABE, which enabled astronomers to measure the position of the stars with unparalleled accuracy.
In medicine they made the first serious studies of drugs and advanced surgery. A number of mathematicians, including Habash al-Hasib (‘he who calculates’), Abul’l-Wafa al-Buzjani, Abu Nasr al-Iraq and Ibn Yunus formulated trigonometry (including all six trig functions [ sin, cosec, cos, sec, tan, and cot ]) at a level far above that introduced by the Greek astronomer-mathematician HIPPARCHUS in the second century BCE.
It is largely through such efforts that Greek ideas were preserved through the DARK AGES.

Eight hundred years before COPERNICUS, a model of the solar system was advanced with the Earth as a planet orbiting the Sun along with other planets.

A few centuries later this idea fell into disfavour with the early Christian Church, which placed mankind at the centre of the universe in a geo-centric model. The alternative teaching would be deemed heresy punishable by death and it would not be until the seventeenth century that the work of GALILEO, KEPLER and NEWTON gave credence to the ideas revitalized by Copernicus in 1543.

It is worth noting that even to-day at least half the named stars in the sky bear Arabic names (Aldebaran and Algol amongst others) and many terms used in astronomy, such as Nadir and Azimuth, are originally Arabic words.

The elaborate observatory established by the Ulugh Begg in Samarkand in the fifteenth century appeared to function with a dictum meant to challenge PTOLEMY’s geocentric picture of the universe sanctioned by the Church in Europe. Arabic scholars had access to the early teachings of ARISTARCHUS, the astronomer from Samos of the third century BCE. (referred to by Copernicus in the forward of an early draft of De Revolutionibus, although omitted from the final copy)

Summarized by the Roman author Lucretius, who wrote ‘On the Nature of the Universe’ in 55 BCE – “The light and heat of the sun; these are composed of minute atoms which, when they are shoved off, lose no time in shooting right across the interspace of air in the direction imparted by the shove”. Accurate for the time, when most people thought that sight was associated with something reaching out from the eye ( EMPEDOCLES ).

Plato wrote of a marriage between the inner light and the outer light.

Euclid worried about the speed with which sight worked. He pointed out that if you close your eyes, then open them again, even the distant stars reappear immediately in your sight, although the influence of sight has had to travel all the way from your eyes to the stars and back again before you could see them.

‘A number of paradoxes; some of which seemed to prove the impossibility of motion’

ZENO

Zeno was a member of the Eleatic school of thought founded by Parmenides. The School held the belief that the underlying nature of the Universe was unvarying and immobile.

In support of Parmenides’ argument, Zeno’s puzzles appeared to show that change and motion were paradoxical and that everything is one – and changeless.

Zeno’s paradoxes were based on the false assumption that space & time are infinitely divisible: that is, the sum of an infinite number of numbers is always infinite. Though they were based on fallacies, the paradoxes remained unsolved for two millennia.

Without the concepts of zero, infinity and the idea of limits, Greek philosophy and mathematics were not equipped to solve the puzzles.

The race between Achilles and the tortoise is not made up of a number of tiny distances but it is continuous until the end.

In the seventeenth century CE the Scottish mathematician James Gregory showed that it is sometimes possible to add infinite terms together to get a finite result – but to do so the terms being added together must approach zero. (This is a necessary, but not sufficient, condition – if the terms go to zero too slowly, then the sum of the terms does not converge to a finite number).
Such a series of numbers is called the convergence series, which occurs when the difference between each number and the one following it becomes smaller throughout the sequence.

The numbers 1, 1/2, 1/4, 1/8, 1/16… are approaching zero as their limit.

When you add up the distance that Achilles runs, with the terms becoming smaller and smaller, each term becomes closer to zero; each term is a step along a journey where the destination is zero.

Since the Greeks rejected the number zero they could not understand that this journey could ever have an end. To them, the terms aren’t approaching anything; the destination does not exist. Instead the Greeks saw the terms as simply getting smaller and smaller.

Using modern definitions, we know that the terms have a limit. The journey has a destination. We now may ask how far away is that destination and how long it will take to get there?

We sum up the distances that Achilles runs:
1 + 1/2 + 1/4 + 1/8 + 1/16… + 1/2^{n} + …
As each step that Achilles takes gets smaller and smaller, it gets closer and closer to zero and the sum of these steps gets closer and closer to 2. How do we know this?
Starting with 2, we subtract the terms of the sum, one by one:

We already know that series has a limit of zero, thus as we subtract the terms from 2 we have nothing left and the limit of the sum 1 + 1/2 + 1/4 + 1/8 + … is 2

The source of the trouble is infinity – Zeno had taken continuous motion and divided it into an infinite number of steps, assuming that the race would continue for ever as the steps get smaller and smaller. The race would never finish in finite time, but with the concept of zero we find a key to solving the puzzle.

Of course, Achilles wins the race.
Achilles takes two steps in catching up with the tortoise, even though he does it in an infinite number of increments.