One of the less comfortable things about those figures at the centre of f57v is that they are drawn dreadfully, and dreadfully in precisely the same way that Kircher’s are in that picture containing ‘Machauter’. While I might speculate on the reasons for that correspondence, I can’t doubt the C-14 dating and can only suppose Kircher had access to (and copied) some image derived from the figure of Matsya (illustrated below). Originally a Creator deity, then a figure who “forewarns Manu about an impending catastrophic flood and orders him to collect all the grains of the world in a boat”, and finally an avatar of Vishnu in the Hindu pantheon, Matsya is not a deity widely known or honoured, and this fact may well prove relevant one of these days.
Of the few sites remaining today dedicated to Matsya, the two most important are at Bet Dwarka in Gujarat ( 22.23°N 68.97°E) on India’s north-west coast, and on the eastern (Coromandel) coast. Vedanarayana Temple stands in what is now a small village named Nagalapuram (13.4000°N 79.7833°E).
Both those sites are near river-mouths and near to the most important of the ancient ports; Barygaza in the west and Arikamedu on the eastern, Coromandal, coast. Higher north on the eastern coast is a site named St.Thomas’ Mount, where tradition held that the saint had been martyred. Revered for more than a millennium by the indigenous Christians, it was to become a centre of Latin Christianity, richly endowed and the centre of a large foreign enclave. (A different Nagalapuram exists in the Ramanathapuram District).
Muziris, which is so well known to western authors lay to the south-west. Pliny said Barygaza was preferable, to avoid pirates, but in Muziris had stood a temple of Augustus, pictured in the Tabula Peutingeriana. And Muziris too was associated with Thomas, and another mount whose name was popularly derived from Thomas’ “peacock” emblem.
Hence – just by the way – my posited identification for the group of plants forming the subject of f.32v: Eastern ‘peacock trees’ – but more on that later).
What has this to do with mathematics and the ‘folded world’ – indeed.
Late in the 5thC AD, an Indian mathematical genius named Aryabhata wrote works in sutra style: a highly condensed style aimed less at prose record than compacting information to enable easier committal to memory and easier recall of details.
It condensed not as an enciphered text would be, but from formal custom just as (in a different way) maths notation is. And he didn’t use numerals but signs, similar to the dual use of alpha-numerics.
Aryabhata, by the way, described the irrational nature of ∏ about thirteen centuries before it was explained to western Europe.
His works were preserved in India by repeated copying (he is still best known in the southern, Tamil regions) and also by translations made into Arabic, and in that way later into medieval Latin.
We may have the idea of the ‘folded world’ because of a mistranslation of his ‘shorthand’ term for the sine.
Aryabhata discussed the concept of sine in his work by the name of ardha-jya, which literally means “half-chord”. For simplicity, people started calling it jya. When Arabic writers translated his works from Sanskrit into Arabic, they referred [to] it as jiba. However, in Arabic writings, vowels are omitted, and it was abbreviated as jb. Later writers substituted it with jaib, meaning “pocket” or “fold (in a garment)”.
(In Arabic, jiba is a meaningless word.) Later in the 12th century, when Gherardo of Cremona translated these writings from Arabic into Latin, he replaced the Arabic jaib with its Latin counterpart, sinus, which means “cove” or “bay”; thence comes the English sine.
Hence – “I like the cut of your jib?” 🙂
Al-Biruni and al-Khwarizmi are among writers in Arabic who evidently knew the Aryabhatiya, which covers arithmetic, algebra, plane trigonometry, and spherical trigonometry. It also contains continued fractions, quadratic equations, sums-of-power series, and a table of sines.
If you imagine these added to the calculations by Eratosthenes, there was not much more that a chartmaker would need in his repertoire save a chart of latitudes and longitudes.
So in this way, again, those regions referenced by the imagery show a pattern consistent with the historical record.
As noted earlier, the style in which additions to f.86v appear to me to have been made accords with the time that the Genoese had contact with Persia, the time when the crossbow reappears in western Europe (chiefly associated with the Genoese), and while in the north, in Trebizond, a Byzantine scholar was melding such sources as Claudius Ptolemy’s tables and the work of al-Khwarizmi.
So I think the ratios given the four radii at the centre of f.57v are most likely meant as ratios for the cartographers’ world.
But whose ratios is the problem, isn’t it?
I’ll come back to this – and to paranatellonta – at some stage.