“…deep down in the heart of piety lurks an insane and lunatic arbitrariness that knows that it has itself produced its God.”
Johannes Climacus (AKA Søren Kierkegaard)
It is claimed that Eratosthenes was known by his peers as Beta, because he was the second-best in the ancient Greek world at just about everything; he devised an algorithmic method for determining prime numbers which marked him out as the second-best mathematician, and his version of the myths about the constellations was some of the second-best poetry ancient Greece had ever heard. His poetry was good, but not so good that anyone ever bothered to write much of it down, so the vast majority of his work has long been lost. However, Eratosthenes is reputed to have been the first person to use the term ‘geography’ and was the first to measure the circumference of the Earth. He did so with an amazing degree of accuracy, although it is unclear exactly how accurate his measurement was because units of measurement were yet to be standardised. Standardised units of measurement were not invented for another two millennia.
Beta made his calculation of the Earth’s circumference without leaving his home city of Alexandria. He was employed as the head of the Library of Alexandria, while there he read that in Swenet, which occupies the site where Aswan is today which sits on the Tropic of Cancer, a vertical column or a pole set plumb in the ground will cast no shadow at noon on the longest day of the year, for at that moment the sun is directly overhead in the sky so that its light will reach the very bottom of the deepest well. Beta knew that the same was not true in Alexandria and he realised that by measuring the angle at which a vertical column casts a shadow at noon on the summer solstice in Alexandria he could ascertain how many degrees of the Earth’s circumference separated the two sites. All that was then needed for Beta to make a calculation of the total circumference of the planet was to know the distance from Alexandria to Swenet.
Beta approximated the distance by asking a travelling merchant how long the journey would take by camel. This is the kind of sloppiness that makes the difference between being first-best and being nicknamed Beta. It also suggests that Eratosthenes’ accurate calculation was an educated-guess benefitting from mutually compensating errors rather than the fruit of the rigorous application of scientific method.
However, for our purposes the accuracy of Eratosthenes' calculation is of secondary interest compared to the fact of his having set out to make the calculation in the first place and its implications for the our conception of the globe.
Around 250 years later, the Roman geographer Pomponius Mela drew a diagram of the planet Earth stratified into distinct zones; freezing polar regions - too cold to inhabit, the temperate area in which the Roman Empire was situated and an equatorial band which would be too hot to navigate across. Mela hypothesised that on the southern side of this equatorial zone there should be an antipodean temperate region mirroring our own which, in theory, could be home to un-contactable races of men. Later still, in the second century, Ptolemy wrote Geographia in which he gave instructions for the drawing up of maps of the known world. When he did so, Ptolemy knew (thanks to Eratosthenes) that the area he was describing represented only about one quarter of the surface of the globe. Eratosthenes’ measurement, to borrow a phrase from the amateur epistemologist Donald Rumsfeld, meant that the three quarters of the globe that were yet to be mapped had moved from being an unknown unknown, to being a known unknown; terra incognita awaiting exploration, conquest and exploitation.
That which can be measured can be ascribed a value, that which can be measured may be sold or speculated upon. Eratosthenes’ measurement of the globe was a catalyst for the beginning of globalisation. It would take until the seventeenth century before trade was sufficiently globalised that it became urgent to solve the problem of a universal unit of measurement, and when the metric system was developed it was as much a symptom of the revolutionary politics of the time as it was a response to necessity. In his 1668 essay Towards a Real Character and a Philosophical Language John Wilkins went further than proposing a universal system of weights and measures; he integrated it within a universal language that would allow merchants, scientists and philosophers from around the world to communicate without their meaning being corrupted by translation or by the fundamental arbitrariness of natural language.
Wilkins was influenced in the construction of his language by a misapprehension that the symbols of Chinese writing make up a form of pasigraphy in which each symbol represents a distinct self-defining concept. In Wilkins’ linguistic scheme everything in the universe is put into one of forty Genera, each Genera represented by a two-letter syllable. These Genera are subdivided into Differences by the addition of a third letter and then subdivided further into Species by the addition of a fourth. By learning the Genera, Differences and species, the speaker of Wilkins’ language may name any animal, mineral or vegetable they might encounter without having to learn what it is it first. Conversely, upon encountering a hitherto unknown word, the speaker may derive an objective definition of whatever it describes.
Wilkins and his contemporaries wished to conceive of a system of measurement that would supersede such arbitrary and inaccurate measures as the acre – which was originally defined as the area that can be tilled by a yoke of oxen within a single day – or the stadia – the unit of distance used by Eratosthenes – although it is unclear whether he adopted the Greek or the Egyptian stadia. The new metric system was to be arrived at scientifically. In Towards a Real Character and a Philosophical Language John Wilkins proposed a unit of length based on the length of a pendulum whose beat is precisely one second, an idea which he borrowed from Christopher Wren. However, the difficulty of accurately measuring one second complicated this apparently simple idea. A proposal made by Gabriel Mouton was adopted instead and, via a series of compromises, one metre came to be defined as one ten-millionth of the length of a meridian running between the North Pole and the equator on a line of longitude running through Paris.
The difficulty of accurately measuring the distance from the pole to the equator, not to mention the misshapen nature of the globe, resulted in the prototype one-metre bar, minted from gold in 1793, being too short by a fraction of a millimetre. Successively more accurate prototype metres have since been made using ever more (supposedly) objective definitions of the unit.
Like Eratosthenes, Jorge Luis Borges worked as a head librarian. It is easy to see similarities between Wilkins’ language and the Dewey Decimal System used in libraries, and it’s not too great a stretch to suppose that the librarian’s concern with the classification of knowledge contributed to Borges’ identification, in a well-known essay, of the central problem with Wilkins’ scheme of universal categorisation and measurement. While the structure of Wilkins’ language is systematic, there is a degree of arbitrariness in the divisions between each of his Genera, Differences and Species. As an example, Borges examines the eighth of Wilkins’ forty Genera; stones:
“Wilkins’ divides them into common (flint, gravel, slate); moderate (marble, amber, coral); precious (pearl, opal); transparent (amethyst, sapphire); and insoluble (coal, fuller’s earth, and arsenic). The ninth category is almost as alarming as the eighth. It reveals that metals can be imperfect (vermillion, quicksilver); artificial (bronze, brass); recremental (filings, rust); and natural (gold, tin, copper). The whale appears in the sixteenth category: it is a viparous, oblong fish…”
Before questioning the categorisation of the whale as a fish, it is worth asking how Wilkins defines the category fish itself.
As soon as any categorisation is attempted, arbitrariness, ambiguity and imprecision inevitably creep in. Borges suggests that Wilkins’ categories are as ridiculous as the cataloguing of all of the world’s animals that he quotes from the apocryphal Chinese encyclopaedia the Heavenly Emporium of Benevolent Knowledge:
“Those that belong to the emperor, Embalmed ones, Those that are trained, Suckling pigs, Mermaids (or Sirens), Fabulous ones, Stray dogs, Those that are included in this classification, Those that tremble as if they were mad, Innumerable ones, Those drawn with a very fine camel hair brush, Et cetera, Those that have just broken the flower vase, Those that, at a distance, resemble flies”
The measurement from which all of the units of the metric system are derived is no less absurd, no less arbitrary, than any of the categories found in the Heavenly Emporium of Benevolent Knowledge:
A unit equal to one ten-millionth of the distance between the North Pole and the equator along a line passing through the capital of France.
In fact, the criteria by which any division of space is defined; a valley, a river, a peninsula, a coastline or a desert, is necessarily arbitrary to some degree. Any atlas is necessarily incomplete, when compiling an atlas it must be accepted that the spaces which its entries describe cannot represent an exhaustive list of the places on the planet. And these spaces can only be described to within a reasonable degree of exactitude and their description can, at best, be anecdotal and refer only to individual cases rather than providing universal definitions. And this will be the case with each of the entries in this atlas, more of which will be added over time.
-The highest point on the planet
-The Parts of the Planet in Which it is Currently Dark
-The Area wherein a Particular Dialect Dominates over Others
-Geographically Distinct Regions whose Names Share Common Origins
-A Three-Dimensional Form with an Infinite Surface Area but a Finite Volume
-A Burial Ground
Olympus Mons is the tallest mountain on any planet in the solar system. Mars has no sea-level against which to measure Olympus Mons, but sea-level is an inadequate base to measure against anyway as it is subject to perpetual change. Olympus Mons is a volcano standing about 21km above the approximate mean height of the surface of Mars. Measuring from the plain at the foot of the volcano to its summit gives a height of around 22km. Using these parameters, the volcano Mauna Kea is the tallest mountain on Earth; 10km from its base on the seabed to its summit.
However, owing to the low profile of Olympus Mons’ slope, the view from the top is unlikely to be anything to write home about. The mountain’s slopes are shallow enough that, if we stood at the summit, the base would be over the horizon. If we were to stand at the top of the tallest mountain in the solar system we would have the impression that we were at the centre of a vast plain. Olympus Mons is not much of a mountain.
The highest point on Earth is the summit of the volcano Chimborazo, when measured from the centre of the planet. Being farthest from the centre, it is the point on the planet where gravity is weakest, so your rucksack will be lighter here than it was when you packed it at home. But it is unlikely you’ll appreciate the effect as the lack of oxygen at this altitude will mean you’re quite a lot weaker.
This is all because Chimborazo sits on the equator and the Earth bulges nearly 43km at the equator compared to the poles. Something Eratosthenes had not way of suspecting when he attempted to make his measurement of the globe. Presumably, Chimborazo is also the point on the planet nearest to the sun at noon and farthest from the sun at midnight.
Sir George Everest was Superintendant of the Great Trigonometric Survey of India between 1823 and 1843. His renowned pedantry and obsession with precision not only meant that he was well-suited to the task and also suggests that he would appreciate it being pointed out that his name is pronounced Eve-rest not Ever-est like the mountain.
The survey was conducted using huge theodolites, each of which required 12 men to carry it. Up to 700 men at a time worked on the survey which lasted from 1818 until 1921, despite the project originally being expected to take 5 years and almost bankrupting the East India Company. The survey measured heights above sea-level, making corrections to their measurements to compensate for the curvature of the Earth, the unevenness of its curvature and even the gravitational pull exerted on their equipment by nearby mountains.
By 1847, Andrew Waugh had taken over as superintendant, however, the survey had reached India’s northern border and could not proceed into Nepal. From Terai, well inside of Indian territory, Waugh made trigonometric measurements of the tallest known peak in the world, Kangchenjung, over the border in Nepal. He went on to measure Peak XV, which was at a distance of around 230km from his position. Many measurements and height estimates later, Waugh calculated the height of Peak XV as 29,000 feet above sea-level. However, he publicly announced a height of 29,002 feet so as to avoid the impression that he had made the figure up.
Waugh suggested that Peak VX should be renamed after his predecessor.
Ptolemy devised his own system for measuring latitude. The system used today gives latitude in degrees of arc; the equator at zero degrees and to poles at 90. Ptolemy’s system expresses latitude with reference to the length of midsummer day; 12 hours at the equator, 24 hours at the pole.
See also; The Globe
(Elements of the following chapter have appeared previously in the book Idioglossia, an art writing glossary)
I have an indistinct memory of visiting the Meadowhall shopping centre on the outskirts of Sheffield. My mental image of it is confused with the image of countless other shopping precincts; some around the edges of London, other strangely futuristic ones in the north of France, uncannily familiar ones in South America, the ubiquitous malls of American television.
According to popular, albeit disputed, folklore, Meadowhall was designed such that had it been a commercial failure, had it not proven an effective economic stimulus, it could have been easily converted into a prison. The evidence against this urban legend is pretty overwhelming, but there is something of the Panopticon in the layout of the centre.
Many years after my encounter with Meadowhall I found myself in an innocuous discussion about the ‘doughnut effect’; a description of the decline of town centres which is perceived to result from the development of out of town shopping and leisure centres, reducing cities to hollowed-out atolls of suburban sprawl. This particular discussion may have concerned the Merry Hill centre near Dudley in the Midlands. But on reflection I cannot remember whether it was Meadowhall or Merry Hill that I had visited, or for that matter, which of the two was being discussed in relation to doughnuts.
The two places are conflated in my memory; Meadowhall in Sheffield and Merry Hill in Dudley. And I am sure that this confusion arises not from them having similar names, and not because both places occupy ‘enterprise zones’ inaugurated on the sites of defunct steelworks or because upon opening they were the largest and second-largest shopping centres in the country. The confusion arises because I have heard locals refer to them by almost the same nickname; Meadow-Hell and Merry Hell respectively.
It is as if a temporary reversal in the confusion of tongues has resulted in a convergence between the local dialects of two separate islands in the Enterprise Zone Archipelago.
Popular local legend has it that Blackheath in South London owes its name to the plague pits, mass graves of victims of the Black Death that supposedly lie beneath its surface. This legend is given credence, of sorts, by the gangs of carrion crows that congregate on the heath by day in their little plague doctor costumes.
However, the legend is fatally undermined when we consider that the name Blackheath is recorded as long ago as the twelfth century, while the Black Death did not reach England until two hundred years later.
Outside of this legend, the name is often considered possibly to be a corruption of bleak-heath or to derive from an Old English descriptive name combining a term meaning dark-coloured earth with the word for heathland. In reality, the topsoil on the heath is light in colour, not dark, and the ground beneath is gravelly. At the edge of the heath there is a series of pits from which this gravel was once extracted and carried down the hill to Ballast Quay on the River Thames. Here, as the name suggests, the gravel was loaded onto ships as ballast. This ballast would be discarded wherever the boats loaded up with cargo. So it is likely that little bits of the heath may be found scattered right across the globe.
Further downstream where the Thames opens into an estuary is the town of Gravesend, which is linked directly to Blackheath by an old Roman road. A modern A-road now follows the route of that Roman road. Another popular local legend has it that the name Gravesend refers to it being either the site of the last recorded case of the Black Death in England or the place from which corpses were taken for mass burial at sea.
As is the case with Blackheath, variations of the name Gravesend predate the Black Death by centuries; appearing as Gravesham in 1086 where it is listed in the Domesday Book, by 1100 it is referred to as Gravesende.
To be fair to these legends, the Black Death had a rate of attrition so great – killing perhaps half of England’s population – that plague pits must have appeared in every town. The Black Death forms as great a part of the country’s national myth as any war or conquest. But while wars are officially glorified in this myth the Black Death is a subterranean presence.
Gravesend’s most notorious real grave is that of the Native American Indian Pocahontas who died there in 1617, having been one of the first natives of the New World to visit the Old. It seems that Pocahontas' presence in London was a marketing stunt pulled by white settlers from Jamestown Virginia. And it is likely she died from exposure to Old-World diseases rather than from a broken heart, as the Disney version of her story suggests she did.
See also; A burial Ground
Zeno’s paradox of Achilles and the Tortoise, formulated in support of the Elean doctrine that no movement is possible and the universe is eternally static and unchanging, has implications so radically opposed to everyday experience that they can be easily dismissed, even if the logic of the paradox is unassailable. There is something more hideous altogether about Gabriel’s Horn; a three-dimensional shape with a finite volume, but an infinite surface area. Yet such a shape exists, on paper at least.
Its outline can be drawn by tracing a line on an x/y axis where y=1/x and x>1...
...and then rotating this line 360 degrees around the x-axis.
Put more straightforwardly it looks like a horn, or a vuvuzela of infinite length which tapers to become infinitesimally narrow and infinitely long.
The idea of the horn’s infinite surface area is conceivable, if not really graspable; Gabriel’s Horn runs on to an infinite length, never ending, getting narrower and narrower. However, understanding how the horn can have a finite volume, despite its infinite surface area, is altogether more difficult.
In order to correctly work out the volume of Gabriel’s Horn it is necessary to imagine cutting it up into pancake-like slices, then calculating the combined volume of these slices...
Adding together the volume of these slices is analogous to adding up the infinite series of fractions 1/2+1/4+1/8+1/16+1/32… then half as much again and again ad infinitum. In this case, just as Achilles can never catch up with the tortoise, the sum of the fractions can never exceed 1, no matter how long we continue adding half as much again.
Where the radius of the wide end of the horn is one unit, its volume can never exceed pi cubic units. Pi is irrational, transcendent, no more graspable than Gabriel’s Horn is playable, but it is finite nonetheless.
Thomas Hobbes reasoned that to understand this shape with finite volume but infinite surface area one need not be a mathematician or geometrician, but rather, one needs to be mad. The infinite can be accepted even when it cannot be fully comprehended, but the finite volume of Gabriel’s Horn is far more vertiginous than any bottomless pit; every iota of confined space enclosing infinitely many infinitely small subdivisions of space. I could be bounded in a nutshell, and count myself a king of infinite space.
Atop the gateposts at the western entrance of St Nicholas’ Church graveyard in Deptford there sits a pair of carved stone skulls resting on crossed bones. A plaque on the outside of the churchyard wall informs us that a church has occupied this site since at least the 12th century, but offers no more information than that.
Locally it is believed that this graveyard has an association with piracy, being the burial place of many executed pirates. This seems perfectly plausible; pirates certainly were hung, or subjected to even more grisly punishments, not far from here.
The tradition associating St Nicolas’ with piracy also holds that the skulls and crossed bones on the gateposts are the original inspiration for the design of the Jolly Roger flag. There seems to be no evidence to back up this idea.
St Olave’s Church in the City of London has a similarly grisly memento-mori above the gates of its small graveyard. This one depicts three skulls grimacing, or maybe laughing. A sign on the outside wall of this graveyard tells us that St Olave’s is one of the only medieval churches to have survived the Great Fire and that Samuel Pepys was buried here in 1703. Furthermore, the sign tells us, there are 365 names of plague victims on the burial register of 1665, and Mother Goose was buried here on September 14th 1586. No further information is given regarding this bizarre final claim.
Back at St Nicolas’ in Deptford, there is another mystery. At the eastern entrance to the graveyard, the far side from the skulls, on the inside of each of the gateposts there are embedded three gold balls in a triangular formation. The only place that this symbol is normally seen is outside a pawnbroker. Its presence here is puzzling. Perhaps this symbol is intended to remind us that life is borrowed, that wealth and status are mere worldly things. In this case, the three gold balls are indicators of the original meaning of the memento-mori; a symbol of piety rather than something gruesome.