The Nuptial Number Revisited

The so-called 'Nuptial Number' is by far Plato's most intractable mathematical problem. It is so dense and so obscure that many commentators, both ancient and modern, have decided that it has no rational solution but is instead an item of satire, not to be taken seriously. Others - including the present author (an admitted obscurantist) - see it as one of the great keys of the Platonic canon, if only it is understood correctly. Like the impossibly obscure formulae of the alchemists, it is the key to the Mysteries. 

It is not the purpose of this post to offer any sort of detailed solution to the 'problem' posed by this most difficult of passages from Plato's Republic; rather its limited purpose is simply to present the passage in question in a clear, accurate translation usefully divided into its component parts. The Greek text is impossibly convoluted and many of the words used are both rare and esoteric. No translation can save the passage from what seems to be deliberate obfuscation on the part of the author. But much of it can be disentangled and made clearer by some explanatory additions and a careful break down of its parts. 

What the infamous 'number' might be and what its significance might be is another matter, but the drift of the passage in itself  - leaving aside the formula of the number - can be straightened out into a largely lucid account. That is what is attempted below. The rendering is based upon those of Benjamin Jowett and James Adam (with copious references to the lexicon of Liddel & Scott. We begin at Republic 545D:

THE PROBLEM: WHAT CAUSES STATES TO DECLINE?

“Come, then,” I said, “and let us try to discern the way a timocracy will develop from of an aristocracy. Or is it the simple and unvarying law that in every form of government disturbance begins among the ruling class itself, when sedition arises among it, but as long as it is at one with itself, disturbance will not occur?”

“Yes, that is so.”

“How, then, Glaucon,” I said, “will disturbance arise in our ideal city, and how will our auxiliaries and rulers fall out and be at odds with one another and with themselves?

INVOCATION TO THE MUSES

Shall we invoke the Muses as Homer does in order to tell us “‘how faction first fell upon them,’” and shall we, like a tragic poet, picture the Muses speaking in an elevated style, as if they are speaking seriously when in fact they are playing with us and teasing us like one child teases another?

“How?”

“In some such fashion as follows:

THE WORDS OF THE MUSES

[Having been invoked in the Homeric manner - albeit ambiguously - the Muse now speaks, and it is through this quasi-Homeric Muse that we are given the Nuptial Number. It is an open question as to how seriously we should take this entire construction, although let us note that Plato uses this "Muse" in other contexts in other dialogues. Many of the 'Platonic myths' are introduced in this way.]

ALL THINGS PASS

It is indeed difficult for a state ideally constituted as we have described to be shaken and disturbed; but since for everything that is born there is also a time of destruction, even a constitution such as ours will not last forever, but it shall surely be dissolved eventually.

This is the manner of its dissolution:

THINGS MOVE IN CYCLES

Not only for plants that grow from the earth but also for animals that live upon it there is a cycle of fertility and barrenness of soul and body as often as the revolutions of their orbs come full circle, in short cycles for the short-lived things and long cycles for long-lived things.

BEGETTING CHILDREN OUT OF SEASON

Concerning human beings, there are also laws of fecund birth and of infertility, and there will come a time when these laws will escape the men you have bred to be the rulers of your city. For all their wisdom, and combining calculation with observation, they will beget children out of season.

A PERIOD COMPREHENDED BY A NUMBER

Now for divine creatures there is a period comprehended by a number that is final and perfect, but for a mortal the number is the first in which multiplications of root by square - when they have attained three distances, with four limits, of that which makes like and unlike and waxes and wanes - have rendered all things commensurable with one another.

THE BASE PRODUCES TWO HARMONIES

The base of this, containing the ratio of four to three, yoked with five, produces two harmonies when increased three times.

1. One of them is equal an equal number of times, so many times a hundred.

2. The other is equal length one way but oblong

- one dimension of a hundred numbers determined by the rational diameters of five diminished by one in each case, or of the irrational by two; the other dimension of a hundred cubes of three.

LORD OF BETTER AND WORSE BIRTHS

The sum of these - this entire geometrical number, a number measuring the earth - is lord of better and worse births.

THE NEGLECTFUL GUARDIANS

When the guardians of your ideal state neglect this and marry brides to bridegrooms out of season children of ill-nature and ill-fortune will be born.

NEGLECT OF THE MUSES

The best of their predecessors will indeed make rulers but these offspring, being unworthy, when they have succeeded to their fathers' offices of power, will begin to neglect us muses, though they are our guardians, and will pay too little heed to music, and then to gymnastics, so that the children will deteriorate and grow up without us.

THE METALLIC RACES

And the rulers who come after them will have little of the guardian in them for testing Hesiod's races and your own - races of gold and silver and copper and iron. And iron will be mixed with silver, and copper will be mixed with gold, and this will engender unlikeness within them, and an unevenness which is disharmonious, which things always create war and enmity wherever they are found. This surely is the pedigree of sedition, wherever it arises. As Homer says, "Of this lineage, look you!" (Iliad 6:211)

CONCLUSION

[Here the Muses finish and we return to the conversation between the interlocutors of the dialogue, namely Socrates and Glaucon.]

"And quite right too," said Glaucon. "We affirm what the Muses say as correct."

"Indeed," I said, "because they are Muses."

* * * 

Yours,

Harper McAlpine Black

Chinese Red - Temple Vermillion

The moment you take even a step into the Sino-Asiatic world - any part of east Asia with a significant Chinese population or under Chinese influence, or else into your nearest Chinatown - you enter a world coloured with red. It is very conspicuous. The Chinese adore the colour red. It features in all of their adornments, both domestic and public. But it is not just any colour red - it is a very particular type of red. Not fire engine red. Not Santa Claus red. Not Red Cross red. Not communist red, either. No. It is Chinese lantern red. Chinese temple red. It is a particular, unmistakeable shade of red usually identified as "vermillion" or else, in the past, as "cinnabar". See the colour square above for an example. 

There is no official definition of the exact shade but in the sample given above "vermillion" has the Hex value #E34234. Any shade of red near to it will pass as "Chinese red" as we will call it here. You will find it used in a thousand different ways. Red lanterns. Red ribbons. Red signs. Red seals. The present author has recently arrived in the Chinese section of George Town on the Prince of Wales Island and this "Chinese red" is on display everywhere. He recently ate at the "Red Garden Food Paradise" which is literally "Chinese red" from top to bottom - red tables, red chairs, red writing, red uniforms on the waiting staff. Everything in this distinctive "Chinese red". 

The standard explanation for the love of this colour among the Chinese is entirely unsatisfactory. We are told, unhelpfully, that the Chinese regard it as "auspicious" and that it brings "good luck." More detailed explanations are equally uninformative. We are told that it "symbolises fire" and this "represents spring" and the "direction south" and is therefore "lucky" or "auspicious" for this reason. Certainly, the Chinese are given to preoccupations of "luck", but surely something more lies behind the ubiqitous use of red, and this particular red. How do we explain that this red - this "vermillion" - is regarded as "auspicious", and also why it is so completely and comprehensively "auspicious" that the Chinese use it so extensively in all contexts great and small? What is it about this red, this particular red, that renders it so important to the Chinese? In the pictures below we see some examples of its many uses:

Tradition lacquerware

Temple entrance with lanterns 

Row of lanterns

Traditional seal (or "chop")

Chinese wedding 

Calligraphy

* * * 

The present author offers the following explanation for this characteristically Chinese phenomenon. It is not difficult to piece together the symbolism of this colour in the Chinese tradition:

Until the development of synthetic alternatives, this particular shade of red was traditionally prepared from 'cinnabar', which is to say from Mercury Sulphide (HgS). Cinnabar is a sulphide of mercury that, when ground into a powder, yields a strong, stable permanent red that can be used in paints and lacquers. Good, stable red colourings are relatively rare in nature, so this preparation - a by-product of mining and metallurgy - was especially valued. 

It was not exclusive to the Chinese, though. Cinnabar (the name comes from Greek but is probably Persian in origin) was known and used in other cultures as well.  We see it used as a red ink in medieval European manuscripts, for example, and as a paint used in the murals of Roman Pompei: 

But the Chinese adopted it as their own. The reason for this is that the Chinese tradition - and especially Taoism - is essentially alchemical and cinnabar, as a metallic essence, is a key ingredient in Taoist alchemy. In the Occident alchemy is, and has always been, a peripheral or 'fringe' tradition. In the Chinese spiritual order it is far more central and mainstream. The colour symbolism of 'Chinese red' and its associations with 'good luck' have a basis in and are to be explained by the significance of cinnabar in Chinese alchemy. 

The primary alchemical significance of cinnabar is this: during the mining of gold the miners might encounter 'veins' of red cinnabar (Mercury sulphide)in the bedrock. Gold and cinnabar are often found together. This is because both gold and mercury are heavy metals and such metals tend to be found in the same geological strata. (For the same reason, arsenic and other heavy metals are often found with gold.) 

Thus cinnabar is associated with gold and in the alchemical mythology of gold mining is often called 'Dragon's blood'. Dragons are believed to store and protect gold in their 'lairs' in the womb of the earth. When miners encounter 'veins' of blood-red pigments running through rocks near and around gold deposits they imagine them to be veins of Dragon's blood. This idea is suggested by the word 'vermillion' too, since it comes from the same root as the word 'worm', and a dragon is a 'worm' in many languages. 'Vermillion' means 'the colour of the dragon/worm'. 

The basic idea here is simple and straightforward. Vermillion - dragon's blood - is "lucky" because it signifies the proximity of gold. When a miner encounters cinnabar (dragon's blood) he is in luck, because he knows there is likely to be gold nearby. When he strikes dragon's blood he has struck gold. 

By extension, this colour is associated with gold and with the auric properties of gold in a general sense. Gold here carries its alchemical significance. It is not merely a precious metal valued in terms of wealth; it also signifies spiritual perfection. Accordingly, the Chinese surround themselves with things the colour of 'dragon's blood' because it points to the perfections of gold. Indeed, as we see in the case of the calligraphy illustrated above, we often find the colour gold with 'Chinese red'. Cinnabar/vermillion/dragon's blood goes with gold in Chinese colour symbolism. You can walk into any Chinese temple and see instances of this. 

By understanding these alchemical associations, and by appreciating the inherently alchemical character of the Chinese tradition, we are in a position to appreciate why this particular colour red is so highly regarded by the Chinese. To a large extent, of course, the traditional connections may be forgotten, and so people will merely regard 'Chinese red' as "lucky" in a superstitious way, but the reasons behind the superstition can still be discerned and understood. In effect, the colour signifies gold, as well as all the things that gold itself signifies, especially the spiritual perfection of the 'Golden Race' and such other parallels. It is remarkable that this metallurgic symbolism has persisted and become so pervasive in the Chinese order. Understanding the symbolism of 'Chinese red' is one of the keys to the entire Chinese tradition. 

Yours,

Harper McAlpine Black

The Tile Patterns of George Town

George Town, on the Prince of Wales Island - or Pe Nang as the Malays call it (which means Land of the Betel Nut) - is possibly the best preserved of any colonial city. A defined area, extending outwards from the Chinese jetties, is a listed heritage zone and consists of superb colonial administrative buildings and extensive rows of terrace houses or what the locals call 'link' houses. Some are very old and many are very well preserved. In amongst them, in addition to these architectural riches, are many splendid Chinese clan temples some of which go back many centuries. As elsewhere, the Chinese and numerous other trading communities prospered here under the comparatively benevelent, tolerant and civilly constructive rule of the British. 

A terrace or 'link' house in the back streets of George Town. 

A striking feature of George Town architecture is the tiled floors, walls and pathways found throughout. This current post is a photographic essay illustrating samples of these distinctive tiles. In many instances, the floors have been redone in the 1920s, and it is largely from that era that these designs come, although some are considerably older. As readers can see, the designs are typically geometrical with those based upon the octagon predominating. Other patterns, less common, are floral. Please click on any image to see a large version. 

Yours,

Harper McAlpine Black

Marie Euphrosyne Spartali Stillman

The photograph above is of the classically beautiful young Marie Euphrosyne Spartali posing as the goddess of memory, Mnemosyne, and was taken by Julia Margaret Cameron in the 1860s. Mrs Cameron was much maligned as a photographer in her day. To contemporary tastes her photography was sloppy and uneven, insufficiently formal, but she found admirers among the Pre-Raphaelites, and it was among that nascent artistic circle that she associated at Little Holland House in Kensington; it was in the salon there that this photograph and others with Miss Spatali as model was taken. Here is another:

Mrs Cameron was born in Calcutta, and Little Holland House was leased by Henry Thoby Prinseps of the artistic Prinsep family, directors of the East India Company. This current blog featured an earlier post on the superb draughtsmanship of James Prinsep, once resident in the sacred city of Benares. (See here.) In this present post we begin by underlining the connection of the Prinseps to the Pre-Raphaelite movement. British India both attracted and gave birth to highly creative and intelligent men and women. Mrs Cameron, hosted by Mr Prinsep at Little Holland House, was among them. The influence of the east via these connections - an orientalist influence, that is to say - was part of the Pre-Raphaelite heritage from the outset. This is sometimes not fully appreciated.

Little Holland House, one of the places where the early circle of the Pre-Rephaelite Brotherhood met. It was demolished after the lease contracted by Mr Prinsep expired. 

In any case, Marie Euphrosyne Spartali, of wealthy Greek Orthodox background, was introduced to the Pre-Raphaelite Brotherhood, its circle and its ideals, through the photography sessions with Mrs Cameron, the exact connection being that Mrs Cameron owned a house next to the Spartali family's vineyard on the Isle of Wight. She took this Miss Spartali to Kensington and introduced her to such figures as Dante Gabriel Rosetti and George Frederic Watts. Smitten with Miss Spartali's Hellenic beauty, these artists eagerly adopted her as a model. She features in many famous works in the Pre-Raphaelite canon. Here she is, for instance, in a work by Rosetti, A Vision of Fiammetta (1878) :

At length, however, she married an older gentleman, also part of the same circle, an American art critic named William James Stillman. The marriage was against her family's wishes, and proved to be difficult, but it furthered her connections to the world of artists and provoked her to seek training in drawing and painting. She was particularly taken by the work of Mr Rosetti and approached him to be her teacher. Rosetti, too busy, declined but recommended she approach Ford Maddox Brown. This she did - once more through the salon at the Prinsep's Little Holland House - and she began her own artistic career. She trained under Mr Brown's tutelage for some ten years. 

This is all by way of introducing her here as one of the present author's favourite Pre-Raphaelites. There are many "lost" Pre-Raphaelites, and it is fashionable these days to lament their neglect - especially the neglect of the females. In the case of the work of Mrs Spartali Stillman the neglect is particularly lamentable, because she was a very fine artist who received scant recognition in her own time or since. When she died she noted in her Will that it seemed odd to make a Will when she had nothing of worth to bequeath. In fact, she left a canon of extremely fine paintings in the Ruskinesque Pre-Raphaelite quasi-Quattrocento style; literary subjects, often neo-medievalist, characterized by complex rather than formulaic compositions, an intensity of colour - as opposed to the gloomy browns of academic painting - and a loving attention to pattern, texture and detail. 

Here is one of her most 'orientalist' works, Woman with Lute. We see the unmistakeable Pre-Raphaelite style adapted to a distinctly orientalist purpose. 

A great many of Mrs Spartali Stillman's work are depictions of single female figures in wistful poses, highly reminiscent of Mr Rosetti's work. Here is a typical example, more medieval Christian and less orientalist in tenor, Cloister Lilies:

 

Possibly her best painting in this genre is the delightful Madonna Pietra degli Scrovigni, from 1884, see below. There is a sense of the mystical in this work, the Madonna clutching a dark crystal ball. Like much of Mrs Spartali Stillman's oeuvre it is done in watercolor and gouache on paper, but with heavy, opaque applications of colour that makes it seem like an oil painting, a method promoted by Edward Burne-Jones and other members of the Pre-Raphaelite circle. It is a method that Mrs Spartali Stillman perfected. 

But it is in larger paintings set in landscapes that we see more of her unique talent. It is in these paintings that she stands apart. For example, see one of her very best paintings, the dramatic, bleak isolation of Antigone, below:

Another beautiful painting captures a scene from Boccaccio's Decameron. From 1889 it is entitled, The Enchanted Garden, or more fully, The Enchanted Garden of Messer Ansaldo, or, more fully still, Messer Ansaldo showing Madonna Dionara his Enchanted Garden. Messer Ansaldo falls in love with Madonna Dionara, a faithfully married woman. In an attempt to woo her - deploying sorcery - he makes his winter garden blossom like spring (though, gallantly, does not dishonour her in the end.) It is likely that Mrs Spartali Stillman's painting was the inspiration for the more famous Enchanted Garden of Mr John Waterhouse from 1916. Mrs Spartali Stillman's work is here below:

Several very charming paintings of a distinctly neo-medievalist tone, all of them set on the grounds of Kelmscott Manor, the home of Mr William and Mrs Jane Morris - another artistic centre like Little Holland House - are among this writer's personal favorites and demonstrate the best aspects of Mrs Spartali Stillman's talent. Morris, and Rosetti and the Pre-Raphaelites, took inspiration from the gothic manor and its organic surrounds. Mrs Spartali Stillman did as well. Her paintings of the manor and its grounds are especially attractive, although they deviate from many of the Pre-Raphaelite norms. They are not literary, for example, or based upon Renaissance models. They are more folkish, more naif. The artist is more herself and less an admirer of Rosetti in these works. Here are four such paintings, all of them splendid:

Kelmscott Manor: Feeding Doves in Kitchen Yard

The Long Walk at Kelmscott Manor

A Lady in the Garden, Kelmscott Manor

From the Field, Kelmscott Manor

In her later career, Mrs Spartali Stillman moved to Italy where her husband was working. There she turned to Tuscan landscapes and other Italian themes, building upon the Quattrocento themes of the Pre-Raphaelite movement. To this writer's tastes, though, her Tuscan work is not so enchanting, although much of it remains in private collections and is rarely available for public display. This, rather than some Victorian conspiracy against female artists, is largely what accounts for the on-going neglect of her work. Unfortunately, almost all discussion of her art is today part of the tiresome oppressed-woman-artist narrative typical of our times. There is much more of interest in her work than what can be seen through the filter of her gender. Quite apart from such contemporary preoccupations, she is a deserving artist with a distinct character who used the Pre-Raphaelite style as a medium for her own unique viewpoint. Readers can be assured that she is not featured here because she is a woman, but because she is good. 


Yours,


Harper McAlpine Black