# Uniformly at Random

## A little more from Titus Groan

Here is another except from Peake’s Titus Groan:

This is a love that equals in its power the love of man for woman and reaches inwards as deeply.  It is the love of a man or of a woman for their world.  For the world of their centre where their lives burn genuinely and with a free flame.

The love of a diver for his world of wavering light.  His world of pearls and tendrils and his breath at his breast.  Born as a plunger into the deeps he is at one with every swarm of lime-green fish, with every coloured sponge.  As he holds himself to the ocean’s faery floor, one hand clasped to a bedded whale’s rib, he is complete and infinite.  Pulse, power and universe sway in his body.  He is in love.

The love of the painter standing alone and staring, staring at the great coloured surface he is making.  Standing with him in the room the rearing canvas stares back with tentative shapes halted in their growth, moving in a new rhythm from floor to ceiling.  The twisted tubes, the fresh paint squeezed and smeared across the dry upon his palette.  The dust beneath the easel.  The paint has edged along the brushes’ handles.  The white light in a northern sky is silent.  The window gapes as he inhales his world.  His world: a rented room, and turpentine.  He moves towards his half-born.  He is in love.

Written by uncudh

December 28, 2008 at 8:00 pm

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## The Cloud Messenger

The poet Kalidasa founded an entire genre of Sanskrit poetry with his lyric poem Meghaduta, or The Cloud Messenger, namely the genre of duta-kavya or messenger poems.  Messenger poems are centered around the theme of separated lovers, where one lover wishes to send a message to his/her beloved.  The poem consists largely of a picturesque description of the scenery through which the messenger will travel, along with a description of the message itself.  In Kalidasa’s The Cloud Messenger, an exiled yaksha (a sort of spirit) requests a passing cloud to take a message to his wife.  Here is one verse (2.64, from the Clay Sanskrit Library translation):

Where, with their various wonders,
they their colorful pictures;
drums are beaten in them to make music,
you have water inside,
they have floors made of jewels;
you are lofty,
their turrets kiss the clouds.

Written by uncudh

December 28, 2008 at 7:51 pm

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In Perilous Realms, Marjorie Burns makes an interesting point concerning the Galadriel/Shelob pairing.  She first notes that Tolkien’s characters often come in contrasting pairs: Gandalf/Saruman, Theoden/Denethor, Frodo/Gollum, etc.  She suggests that Galadriel/Shelob represents another such pairing.  Furthermore, Frodo’s confrontation with Shelob can be viewed as actually being a confrontation between Galadriel and Shelob, with Galadriel’s phial being in some sense her proxy.  Furthermore, since the light of the phial comes originally from one of the Silmarils, a connection is made to the earlier tale of the theft of the Silmarils by Morgoth and Ungoliant, the spider ancestress of Shelob.

It is not until The Lord of The Rings that Tolkien allows his lady-and-the-spider drama to unfold, and even then he fulfils the drama only symbolically. In this, however, Tolkien’s instincts are correct. For one, by replacing Galadriel with a symbol of her power, Tolkien greatly increases the significance of the confrontation with Shelob. Galadriel’s phial, in the words of Christopher Tolkien, is a ‘huge power, a veritable star in the darkness’ (IX, 13), and its history extends well beyond the boundaries of Middle-earth and well beyond the time period or characters we meet in The Lord of The Rings. The phial that holds Shelob at bay contains the light of the Silmaril that adorned Eärendil’s ship before Elbereth placed it in the sky as a star; and the light that came from this remaining Silmaril came first from the Two Trees of Valinor, the trees that Ungoliant destroyed by drinking up their light.  The story of Galadriel’s phial thus stretches from the days of creation to the end of the Third Age, adding not only far greater meaning to the confrontation with Shelob but bringing together forces from the highest level of Valinor to the hobbits of Middle-earth.

Written by uncudh

December 20, 2008 at 8:51 pm

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## Mariana

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From Tennyson’s “Mariana”:

With blackest moss the flower-plots
Were thickly crusted, one and all:
The rusted nails fell from the knots
That held the pear to the gable-wall.
The broken sheds look’d sad and strange:
Weeded and worn the ancient thatch
Upon the lonely moated grange.

She only said, ‘My life is dreary,
He cometh not,’ she said;
She said, ‘I am aweary, aweary,
I would that I were dead!’

Written by uncudh

December 20, 2008 at 8:35 pm

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## A curious proof of Wilson’s Theorem

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Recall that Wilson’s Theorem states that $p$ is a prime number if and only if $(p-1)! \equiv -1 \pmod p$. Wilson did not prove this statement; the first proof is due to Lagrange in 1773. Here we give a proof due to Thue from 1893 that uses the calculus of finite differences. Fortunately we have done most of the work previously.

Suppose $p$ is prime. Recall that

$\Delta^{(r)}(f)(x) = \sum_{j=0}^r (-1)^{r-j} {r \choose j} f(x+j).$

We also showed that $\Delta^{(r-1)}(x^r) = r!x + d$ for some constant $d$, so that $\Delta^{(r)}(x^r) = r!$. Let $f(x) = x^{p-1}$. It follows then that $\Delta^{(p-1)}(f)(0) = (p-1)!$. On the other hand, for $r = 1,2,\ldots,p-2$ we have

$\Delta^{(r)}(f)(1) = \sum_{j=0}^r (-1)^{r-j} {r \choose j} (1+j)^{p-1}.$

However, by Fermat’s Little Theorem, we have $(1+j)^{p-1} \equiv 1 \pmod p$, so

$\Delta^{(r)}(f)(1) \equiv \sum_{j=0}^r (-1)^{r-j} {r \choose j} \equiv (1-1)^r \equiv 0 \pmod p,$

where we have applied the Binomial Theorem. Now

$\Delta^{(p-1)}(f)(0) = \Delta^{(p-2)}(f)(1) - \Delta^{(p-3)}(f)(1) + \cdots - \Delta^2(f)(1) + \Delta(f)(1) - \Delta(f)(0),$

but by our previous observation, all of the terms on the right hand side are congruent to $0 \pmod p$, except the last, $-\Delta(f)(0)$. We thus have

$\Delta^{(p-1)}(f)(0) \equiv -\Delta(f)(0) \equiv f(0) - f(1) \equiv 0^{p-1} - 1^{p-1} \equiv -1 \pmod p$.

Thus, $(p-1)! \equiv -1 \pmod p$, as required.

Written by uncudh

December 20, 2008 at 3:22 am

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## Motivations for Iroquois warfare

Hunt, in his 1940 book, The Wars of the Iroquois: A Study in Intertribal Trade Relations, described the Iroquois Wars as being fought principally so that the Iroquois could establish themselves as middlemen in the fur trade. For instance, according to this interpretation, since the Hurons had established themselves in this role in the trade between the nations living further west in the Great Lakes regions and the French at Montreal, the Iroquois made war on the Hurons in the 1640’s in order to drive them out of this preferential position and divert the fur trade southward, through the Iroquois country, to the Dutch (later English) at Albany. In this way the Iroquois could control the west-east flow of furs. Hunt presents these economic forces as the central impulse driving Iroquois policy.

As a non-expert, after reading Hunt’s account his arguments seemed rather convincing to me. However, many later historians have severely criticised his explanations of Iroquois behaviour. For example, Brandao’s book, ‘Your fyre shall burn no more’: Iroquois Policy towards New France and its Native Allies to 1701, rejects such economic motivations and instead presents the thesis that the principal motivations behind Iroquois warfare were cultural in nature. For instance, in his view, motives for Iroquois warfare included the need to acquire captives for adoption into Iroquois society in order to recoup losses due to epidemics or previous military conflict, or the desire of Iroquois men to acquire prestige in the only way available to them, namely by displaying prowess in battle.

Brandao therefore takes an almost diametrically opposite point of view to that of Hunt. Again, as a non-expert, after reading Brandao I found his arguments to be rather convincing. However, other historians do not subscribe to his interpretation either. Trigger writes in the preface to the 2000 printing of The Children of Aataentsic:

Very different in its objectives is Jose Brandao’s rancorous and polemical book […] Like his mentor, William J. Eccles, Brandao accuses the majority of Iroquoian ethnohistorians, myself included, of being economic reductionists who underestimate the resilience and importance of traditional Iroquoian culture and fail to interpret Iroquoian behaviour in ways that would have made sense to these peoples. A wide range of materialist approaches are caricatured as being based on the false belief that during the period of the fur trade the Iroquois were motivated by a universal desire for economic gain. While it is true, as I observed long ago, that the economists George Hunt and Harold Innis paid little attention to indigenous beliefs, Brandao’s claims seem based on biased and selective readings of even these scholars. He certainly misrepresents the more nuanced, multifaceted, and culturally informed arguments of most anthropologists and historians who have written about Iroquoian groups in recent decades.

Other interpretations of the Iroquois Wars are out there as well: Jennings suggests a desire on the part of the Iroquois to acquire hegemony over the nations of the Ohio, Susquehanna, and Hudson River valleys, etc.

Written by uncudh

December 10, 2008 at 9:50 pm

## Aldarion and Erendis

One of my favourite works by Tolkien is the tale of “Aldarion and Erendis” from Unfinished Tales.  It is notable for several reasons.  It is the only story (i.e., not counting annals and other such records) of Numenor to survive the downfall.  It is also quite unique in content among Tolkien’s writings.  It is not a story of a difficult quest to destroy a great evil, nor is it a tale of heroes slaying monsters, or of spectacular battles between huge armies, but it is instead the tragic story of the failure of a marriage.

Aldarion, heir to throne of Numenor, had been drawn to the sea from an early age.  Seafaring was his passion, and he sailed forth on long and frequent sea voyages between Numenor to Middle-earth, to the great displeasure of his father.  He meets Erendis, a woman of high birth, and the two fall in love.  After some time, Aldarion courts her in earnest and the two are betrothed.  During the long course of their betrothal, and subsequently during their marriage, Aldarion continued to set out to sea, often leading to absences of many years.  This leads to the increasing estrangement of the couple.  Eventually, Erendis leaves their home, taking her daughter Ancalime with her.  When Aldarion returns from sea and seeks out his wife and daughter, they have a rather unpleasant exchange:

Later he would summon Erendis to bring his daughter to Armenelos, and not have dealings with her upon her own ground.  But as he went out towards the doors Erendis came forward.  She had not lain in bed that night, and she stood before him on the threshold.

‘You leave more promptly than you came, my lord,’ she said.  ‘I hope that (being a mariner) you have not found this house of women irksome already, to go thus before your business is done.  Indeed what business brought you hither?  May I learn it before you leave?’

‘I was told in Armenelos that my wife was here, and had removed my daughter hither,’ he answered.  ‘As to the wife I am mistaken, it seems, but have I not a daughter?’

‘You had one some years ago,’ she said.  ‘But my daughter has not yet risen.’

‘Then let her rise, while I go for my horse,’ said Aldarion.

The tale is unfortunately incomplete; the latter portion consists of scattered notes mostly concerning Aldarion’s daughter Ancalime, who would become, as his only child, his heir, and eventually Queen of Numenor.

Written by uncudh

December 10, 2008 at 6:58 pm

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