lander

Aswan, at approximately 24 degrees north latitude, lies very near the Tropic of Cancer (23°27′ N), and at this latitude, near the time of the summer solstice, the Sun is (roughly) 90 degrees above the horizon at noon...

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Around 245 BC, when Eratosthenes was in his 30s, he was working as a librarian in the famous Library of Alexandria in Egypt.

He read about a water well in the city of Syene (modern-day Aswan in southern Egypt).

At midday every summer solstice, the sun would shine directly down into the well, illuminating the water at the bottom - but casting no shadow on the walls of the well.

It meant the sun sat directly above Syene at that exact moment.

So Eratosthenes wondered, if he stuck a pole in the ground in Alexandria at that same moment, would it cast a shadow?

And it turns out it did.

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Since he knew the height of the pole, and the length of the shadow it cast, it meant he knew the lengths of two sides of a right-angled triangle.

That meant he could figure out the length of the third side of the triangle, and he could also figure out the angle at the top of the pole, between the sunbeam and the pole itself.

It was 7.2 degrees.

Therefore, he knew the sun was hitting Alexandria at an angle of 7.2 degrees (precisely) at midday on the summer solstice.

And that left him with one final measurement.

To figure out the circumference of the earth, he needed to somehow measure the distance between Alexandria and Syene.

So he asked someone (or a team of people) to walk it.

Those people were called "bematists", professional surveyors who were trained to measure vast distances extremely accurately by pacing the distance.

They estimated the distance between the two cities was roughly 5,000 stadia (or 800 kilometres).

And that was everything Eratosthenes needed.

He had all the ingredients to calculate the circumference of the earth.

https://www.abc.net.au/news/2021-09-05/eratosthenes-measured-circumference-of-the-earth-2200-years-ago/100434560

A few centuries later and the earth was still flat for some...

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Amazingly, a lot of the measurements in Stadia (paces), length of poles and angles would have had a significant element of statistical error and sensible approximation. Even our conversion of Stadia to kilometres has to be done within an approximated bracketed value. The latitude of Syene (modern-day Aswan) is also out by half a degree (33’) on the solstice. But we also need to take into account is Aswan exactly where Syene was? How far from Syene, the well was, considering that one minute of arc on the surface of the planet is equivalent to one nautical mile.

At sea level one minute of arc along the equator equals exactly one geographical mile along the Earth's equator or (approximately) one nautical mile (1,852 metres; 1.151 miles). A second of arc, one sixtieth of this amount, is roughly 30 metres (98 feet).

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So as ABC business reporter Gareth Hutchens explains Eratosthenes’ discovery, we need to make a few more measurements beyond the fictitious biblical cubits, used to build a fictitious ark by a fictitious character, Noah… in regard to a non-biblical flood, namely the big melt at the end of the last Ice Age...

How the kilometre was defined...

In the early ninth century, when much of “France" was part of the Holy Roman Empire, units of measure had been standardised by Emperor Charlemagne. He had introduced standard units of measure for length and for mass throughout his empire. He obviously did not want to be short-changed by anyone. As the empire disintegrated into separate nations, including France, these standards diverged. In England the Magna Carta (1215) had stipulated that "There shall be standard measures of wine, ale, and corn, throughout the kingdom. There shall also be a standard width of dyed cloth, russet, and haberject, namely two ells within the selvedges. Weights are to be standardised similarly.”

Imagine being a merchant of stuff, you needed a degree in weights and measures to know which stick to use for what length of whatever sold. Corruption of sticks and weights were thus common, keeping inspectors of scales and sticks very busy. At least a dozen egg was a dozen egg, but as we know we can buy 600, 700 or 800 grams of eggs in a dozen (not a metric unit) carton. And I’m sure that a 600 grams of eggs is the minimum allowed for selling 600 grams of eggs. There could be 621 grams of eggs in the carton. Hens do not make exact 50 gram eggs each.

During the early medieval era, Roman numerals were used in Europe, while the Arabs represented numbers using the Hindu numeral system — a notation that used ten symbols. In about 1202, Fibonacci published his book Liber Abaci (Book of Calculation) which introduced the concept of positional notation into Europe. These symbols evolved into the numerals "0", "1", "2" etc. At that time, there was controversy regarding the difference between rational numbers (which can be divided into definite portions — say one tart divided in six serves) and irrational numbers (which divides into indefinite portions — say π = 3,14106…. etc) and there was no consistency in the way in which decimal fractions were represented.

Simon Stevin is credited with introducing the decimal system into general use, in Europe. In 1586, he published a small pamphlet called De Thiende ("the tenth") which historians credit as being the basis of modern notation for decimal fractions. Stevin felt that this innovation was so significant that he declared the universal introduction of decimal coinage, measures, and weights to be merely a question of time.

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In 1790, a proposal by the French was made to Britain and the United States, to establish a uniform measure of length. A metre based on the period of a pendulum with a beat of one second, was defeated in the British Parliament and United States Congress. The contested issue was to define the latitude, since gravitational acceleration, and therefore the length of the pendulum, varies with latitude. Each party wanted a definition according to a major latitude passing through their own country. 

The direct consequences of the failure of this proposal were the French unilateral development and deployment of the metric system and its spread by trade to the continent; the British adoption of the Imperial System of Measures throughout the realm in 1824; and the United States' retention of the British Common System of Measures in place at the time of the independence of the colonies. This was the differing positions that continued for nearly the next 200 years. We know. We still measure oil in US barrels.

Remember:  NASA lost its $125-million Mars Climate Lander/Orbiter because spacecraft engineers failed to convert from English to metric measurements when exchanging vital data before the craft was launched, space agency officials said Thursday.

(OCT. 1, 1999 12 AM PT). See hopeful artist's impression at top

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Soon after the French Revolution, possibly in order to eliminate the “Pied-du-Roi” unit (length of the king’s foot) measurement of stuff (Revolutionary sarcasm allowed here), In 1790, a panel of five leading French scientists was appointed by the Académie des sciences to investigate weights and measures. They were Jean-Charles de Borda, Joseph-Louis Lagrange, Pierre-Simon Laplace, Gaspard Monge and Nicolas de Condorcet. We’ve studied some of these extraordinary thinkers before on this site. Our modern engineering and sciences from relativity to quantum mechanics are still in the grips of these giants’ equations — to which we should also add Cardano's.

Over the following year, the panel, after studying various alternatives, made a series of recommendations regarding a new system of weights and measures, including that it should be decimal, that the unit of length should be based on a fractional arc of a quadrant of the Earth's meridian, and that the unit of weight should be that of a cube of water whose dimension was a decimal fraction of the unit of length. The proposals were accepted by the French Assembly on 30 March 1791.

This in itself is remarkable. It shows an assembly of politicians for which the scientific and system accuracy was primordial, considering the morons who still argue about the shape of the earth in the Australian parliament and fading curtains in Queensland. 

 Following official acceptance, the Académie des Sciences had to calculate and implement the new system. The Académie broke the tasks into five operations, allocating each part to a separate working group:

     • Measuring the difference in latitude between Dunkirk and Barcelona and triangulating between them

     • Measuring the baselines used for the survey

     • Verifying the length of the second (timeframe) pendulum at 45° latitude.

     • Verifying the weight in a vacuum of a given volume of distilled water.

     • Publishing conversion tables relating the new units of measure to the various existing units of measure.

The panel decided that the new measure of length should be equal to one ten-millionth of the distance from the North Pole to the Equator (the quadrant of the Earth's circumference), measured along the meridian passing through Paris.

Using Jean Picard's survey of 1670 and Jacques Cassini's survey of 1718, a provisional value of 443.44 "lignes" was assigned to the metre which, in turn, defined the other units of measure. There are 12 lignes to one French inch (pouce). The standardised conversion for a ligne is 2.2558291 mm (1 mm = 0.443296 ligne), and it is abbreviated with the letter L or represented by the triple prime (‴). It was not a question of discovering “new” kinds of measurements, but of refining the definitions of measuring stuff in relation to the old ways of measuring, with unified standards. Not as easy as it looks.

The task of surveying the meridian arc, which was estimated to take two years, fell to Pierre Méchain and Jean-Baptiste Delambre, while the commission ordered a series of platinum bars to be made based on the provisional metre. When the final result was known, the bar whose length was closest to the meridional definition of the metre would be selected. 

Surveying the meridian took more than six years (1792–1798) with delays caused not only by technical difficulties but also by the aftermath of the Revolution. Apart from the obvious nationalistic considerations, the Paris meridian was also a sound choice for practical scientific reasons: a portion of the quadrant from Dunkirk to Barcelona (about 1000 km, or one-tenth of the total) could be surveyed with start- and end-points at sea level, and that portion was roughly in the middle of the quadrant, where the effects of the Earth's oblateness were expected to be the largest.

The project was split into two parts: the northern section of 742.7 km from the Belfry, Dunkirk to Rodez Cathedral was surveyed by Delambre and the southern section of 333.0 km from Rodez to the Montjuïc Fortress, Barcelona was surveyed by Méchain.

Delambre used a baseline of about 10 km in length along a straight road, located close to Melun. In an operation taking six weeks, the baseline was accurately measured using four platinum rods, each of length two toises (about 3.9 m). Thereafter he used, where possible, the triangulation points used by Cassini in his 1744 survey of France. 

Méchain's baseline, of a similar length, and also on a straight section of road was in the Perpignan area. Although Méchain's sector was half the length of Delambre, it included the Pyrenees as a non-surveyed parts of Spain. After the two surveyors met, each computed the other's baseline in order to cross-check their results and they then recomputed the metre as 443.296 lignes, notably shorter than the 1795 provisional value of 443.44 lignes.

On 15 November 1798 Delambre and Méchain returned to Paris with their data, having completed the survey. The final value of the mètre was defined in 1799 as the computed value from the survey.

It soon became apparent that Méchain and Delambre's result (443.296 lignes) was slightly too short for the meridional definition of the metre. Méchain had made a small error measuring the latitude of Barcelona, so he remeasured it, but kept the second set of measurements secret. The official metre was thus too short, but we won't tell anyone... because it does not matter — as long as the same stick is used by everyone.

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The rest is history...

Note: Using the original metre which is too short from its original intent due to Méchain's miscalculation, the equatorial circumference of Earth is thus 40,075 km and the meridional circumference (pole to pole as measured by the French) is 40,008 km. Our planet's shape, caused by the flattening at the poles, is called an oblate spheroid.

IMPORTANT NOTE: should you live between the tropic of Capricorn and the Tropic of Cancer, there will be one day (actually two days) of the year when at midday, the shadows will be "from a vertical sun" like in Syene. For the Equator, this day will be at the Equinox(es). 

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