The Lumber Room

"Consign them to dust and damp by way of preserving them"

The Indian theory of aesthetic appreciation (rasa)

leave a comment »

Here’s a great, simple write-up aimed at a Western audience, from the Clay Sanskrit Library edition of Kālidāsa’s The Recognition of Shakuntala, written by Somadeva Vasudeva:

Imagine that you find yourself going to see a performance of “Romeo and Juliet.” You are in the right mood for the play, no mundane worries preoccupy your mind, you have agreeable company, and the theatre, the stage, the director and the actors are all excellent—capable of doing justice to a great play. Your seat in the theatre is comfortable and gives an unobstructed view.

The play begins and you find yourself drawn into the world Shakespeare is sketching. The involvement deepens to an immersion where the ordinary, everyday world dims and fades from the center of attention, you begin to understand and even share the feelings of the characters on stage—under ideal conditions you might reach a stage where you begin to participate in some strange way in the love being evoked.

Now, if at that moment you were to ask yourself: “Whose love is this?” a paradox arises.

It cannot be Romeo’s love for Juliet, nor Juliet’s love for Romeo, for they are fictional characters. It cannot be the actors’, for in reality they may despise one another. It cannot be your own love, for you cannot love a fictional character and know nothing about the actors’ real personalities (they are veiled by the role they assume), and, for the same reasons, it cannot be the actors’ love for either you or the fictional characters. So it is a peculiar, almost abstract love without immediate referent or context.

A Sanskrit aesthete would explain to you that you are at that moment “relishing” (āsvādana) your own “fundamental emotional state” (sthāyi-bhāva) called “passion” (rati) which has been “decontextualised” (sādhāraṇīkṛta) by the operation of “sympathetic resonance” (hṛdayasaṃvāda) and heightened to become transformed into an “aesthetic sentiment” (rasa) called the “erotic sentiment” (śṛṅgāra).

This “aesthetic sentiment” is a paradoxical and ephemeral thing that can be evoked by the play but is not exactly caused by it, for many spectators may have felt nothing at all during the same performance. You yourself, seeing it again next week, under the same circumstances, might experience nothing. It is, moreover, something that cannot be adequately explained through analytic terms, the only proof for its existence is its direct, personal experience.


It is, moreover, a blissful experience. The fact that sensitive readers often weep while reading poetry does not mean that they are suffering, rather the tenderness of the work has succeeded in melting the contraction of their minds or hearts.

The non-ordinary nature of such aesthetic sentiments makes it possible for the spectator or reader to derive a pleasurable experience even from what in ordinary life would be causes of grief.

The Indian scholarly tradition has a lot more, including some very thoughtful deliberation and perceptive observation, but it seems good to start a discussion of rasa with an example like this, than to start with the technical details.

[Another good start may be via film. See for instance:
How to Watch a Hindi Film: The Example of Kuch Kuch Hota Hai by Sam Joshi, published in Education About Asia, Volume 9, Number 1 (Spring 2004).
and perhaps (and if you have a lot of time):
Is There an Indian Way of Filmmaking? by Philip Lutgendorf, published in International Journal of Hindu Studies, Vol. 10, No. 3 (Dec., 2006), pp. 227-256.
Previously on this blog: On songs in Bollywood]

Written by S

Fri, 2014-06-06 at 23:46:41

The rest is commentary

leave a comment »

Famous verses appear in many variants. Thanks to Google, it is easy to find many of them. For “paropakāraḥ puṇyāya, pāpāya parapīḍanam”, Google throws up a lot of variants for the first half.

The Vikramacarita has:

śrūyatāṃ dharmasarvasvaṃ, yad uktaṃ śāstrakoṭibhiḥ /
paropakāraḥ puṇyāya, pāpāya parapīḍanam

Other variants are:

saṅkṣepāt kathyate dharmo janāḥ kiṃ vistareṇa vaḥ |
paropakāraḥ puṇyāya pāpāya para-pīḍanam ||Panc_3.103||


aṣṭādaśapurāṇeṣu vyāsasya vacanadvayam /
paropakāraḥ puṇyāya pāpāya parapīḍanam //


ślokārdhena pravakṣyāmi yaduktaṃ grantha-koṭibhiḥ
paropakāraḥ puṇyāya pāpāya parapīḍanam

Going by the first line gives other verses:

śrūyatāṃ dharmasarvasvaṃ śrutvā caivāvadhāryatām | (or caiva vicāryatām ।)
ātmanaḥ pratikūlāni pareṣāṃ na samācaret ||

[Cāṇakya-nīti, Pañcatantra, Subhāṣitāvalī etc.]

See also:

prāṇā yathātmano ‘bhīṣṭā bhūtānām api te tathā |
ātmaupamyena gantavyaṃ buddhimadbhir mahātmabhiḥ

तस्माद्धर्मप्रधानेन भवितव्यं यतात्मना ।
तथा च सर्वभूतेषु वर्तितव्यं यथात्मनि ॥ Mahābhārata Shānti-Parva 167:9

05,039.057a na tatparasya saṃdadhyāt pratikūlaṃ yadātmanaḥ
05,039.057b*0238_01 ātmanaḥ pratikūlāni vijānan na samācaret
05,039.057c saṃgraheṇaiṣa dharmaḥ syāt kāmād anyaḥ pravartate

As Hillel says, the rest is commentary.

For (some) commentary, go here.

Written by S

Fri, 2014-06-06 at 23:38:30

Posted in quotes, sanskrit, unfinished

Prefatory apprehension

leave a comment »

Robert Recorde’s 1557 book is noted for being the first to introduce the equals sign =, and is titled:

The Whetstone of Witte: whiche is the seconde parte of Arithmeteke: containing the extraction of rootes; the cossike practise, with the rule of equation; and the workes of Surde Nombers.

Its title page (see, see also the full book at contains this verse:

Original spelling

Though many ſtones doe beare greate price,
whetſtone is for exerſice
As neadefull, and in woorke as ſtraunge:
Dulle thinges and harde it will ſo chaunge,
And make them ſharpe, to right good vſe:
All arteſmen knowe, thei can not chuſe,
But uſe his helpe: yet as men ſee,
Noe ſharpeneſſe ſemeth in it to bee.

The grounde of artes did brede this ſtone:
His vſe is greate, and moare then one.
Here if you lift your wittes to whette,
Moche ſharpeneſſe thereby ſhall you gette.
Dulle wittes hereby doe greately mende,
Sharpe wittes are fined to their fulle ende.
Now proue, and praiſe, as you doe finde,
And to your ſelf be not vnkinde.

Modern spelling

Though many stones do bear great price,
The whetstone is for exercise
As needful, and in work as strange:
Dull things and hard it will so change
And make them sharp, to right good use:
All artsmen know they cannot choose
But use his help; yet as men see,
No sharpness seemeth in it to be.

The ground of arts did breed this stone;
His use is great, and more than one.
Here if you lift your wits to whet,
Much sharpness thereby shall you get.
Dull wits hereby do greatly mend,
Sharp wits are fined to their full end.
Now prove and praise as you do find,
And to yourself be not unkind.

Apparently the full title contains a pun (see “the cossike practise” in the title refers to algebra, as the Latin cosa apparently meaning “a thing” was used to stand for an unknown, abbreviated to cos — but the Latin word cos itself means a grindstone.

The author again reminds readers not to blame his book, at the end of his preface:

To the curiouſe ſcanner.

If you ought finde, as ſome men maie,
That you can mende, I ſhall you praie,
To take ſome paine ſo grace maie ſende,
This worke to growe to perfecte ende.

But if you mende not that you blame,
I winne the praiſe, and you the ſhame.
Therfore be wiſe, and learne before,
Sith ſlaunder hurtes it ſelf moſte ſore.

Authors are either anxious about how their book is received, or make sure to be pointedly uncaring.

Sir Arthur Conan Doyle, in a mostly forgettable volume of poetry (Songs of the Road, 1911), begins:

If it were not for the hillocks
   You’d think little of the hills;
The rivers would seem tiny
   If it were not for the rills.
If you never saw the brushwood
   You would under-rate the trees;
And so you see the purpose
   Of such little rhymes as these.

Kālidāsa of course begins his Raghuvaṃśa with a grand disclaimer:

kva sūryaprabhavo vaṃśaḥ kva cālpaviṣayā matiḥ /
titīrṣur dustaram mohād uḍupenāsmi sāgaram // Ragh_1.2 //

mandaḥ kaviyaśaḥ prārthī gamiṣyāmy upahāsyatām /
prāṃśulabhye phale lobhād udbāhur iva vāmanaḥ // Ragh_1.3 //

atha vā kṛtavāgdvāre vaṃśe ‘smin pūrvasūribhiḥ /
maṇau vajrasamutkīrṇe sūtrasyevāsti me gatiḥ // Ragh_1.4 //

But the most nonchalant I’ve seen, thanks to Dr. Ganesh, is this gīti by Śrīkṛṣṇa Brahmatantra Yatīndra of the Parakāla Maṭha, Mysore:

nindatu vā nandatu vā
mandamanīṣā niśamya kṛtim etām
harṣaṃ vā marṣaṃ vā
sarṣapamātram api naiva vindema

Screw you guys. :-)

Written by S

Wed, 2014-05-28 at 23:56:11

Posted in history, mathematics

Big O() notation: a couple of sources

with 2 comments

This post contains, just for future reference, a couple of primary sources relevant to the O (“Big O”) notation:

  1. Some introductory words from Asymptotic Methods in Analysis by de Bruijn
  2. An letter from Donald Knuth on an approach to teaching calculus using this notation.

Read the rest of this entry »

Written by S

Thu, 2014-03-13 at 16:33:20

Visualizing product of permutations

with 4 comments

A simple pedagogical trick that may come in handy: represent a permutation \sigma using arrows (curved lines) from k to \sigma(k) for each k. Then, the product of two permutations can be represented by just putting the two corresponding figures (sets of arrows) one below the other, and following the arrows.

Representing permutations and products of permutations.

Representing permutations and products of permutations.

The figure is from an article called Symmetries by Alain Connes, found via the Wikipedia article on Morley’s trisector theorem (something entirely unrelated to permutations, but the article covers both of them and more).

I’m thinking how one might write a program to actually draw these: if we decide that the “height” of the figure is some h, then each arrow needs to go from some (k, 0) to (\sigma(k), h) (using here the usual screen convention of x coordinate increasing from left to right, and y coordinate increasing from top to bottom). Further, each curve needs to have vertical slope at its two endpoints, so that successive curves can line up smoothly. The constraint on starting point, ending point, and directions at the endpoints defines almost a quadratic Bezier curve, except that here the two directions are parallel. So it’s somewhere between a quadratic and the (usual) cubic Bezier curve, which is given by the start point, end point, and derivatives at the start and end point. (Here we only care about the direction of the derivative; we can pick some arbitrary magnitude to fix the curve: the larger we pick, the more smooth it will look at the ends, at the cost of smoothness in the interior.)

Even knowing the curve, how do we generate an image?

Written by S

Thu, 2014-03-06 at 23:15:44

The letter in Roister Doister

leave a comment »

An early English example of resegmentation for change of meaning: Merygreeke’s letter to the Dame Custance, from Ralph Roister Doister (c. 1553), the first comedy to be written in the English language. [Aside: Shakespeare born in 1564 started had written all his plays before 1613, only 60 years from then.]

The example also shows that enjambment (no pause at end of line) did indeed exist in early English as well.

Scene 4

Sweete mistresse where as I loue you nothing at all,
Regarding your substance and richesse chiefe of all,
For your personage, beautie, demeanour and wit,
I commende me vnto you neuer a whit.
Sorie to heare report of your good welfare.
For (as I heare say) suche your conditions are,
That ye be worthie fauour of no liuing man,
To be abhorred of euery honest man.
To be taken for a woman enclined to vice.
Nothing at all to Vertue gyuing hir due price.
Whersore concerning mariage, ye are thought
Suche a fine Paragon, as nere honest man bought.
And nowe by these presentes I do you aduertise
That I am minded to marrie you in no wise.
For your goodes and substance, I coulde bee content
To take you as ye are. If ye mynde to bee my wyfe,
Ye shall be assured for the tyme of my lyfe,
I will keepe ye ryght well, from good rayment and fare,
Ye shall not be kepte but in sorowe and care.
Ye shall in no wyse lyue at your owne libertie,
Doe and say what ye lust, ye shall neuer please me,
But when ye are mery, I will be all sadde,
When ye are sory, I will be very gladde.
When ye seeke your heartes ease, I will be vnkinde,
At no tyme, in me shall ye muche gentlenesse finde.
But all things contrary to your will and minde,
Shall be done: otherwise I wyll not be behinde
To speake. And as for all them that woulde do you wrong
I will so helpe and mainteyne, ye shall not lyue long.
Nor any foolishe dolte, shall cumbre you but I.
Thus good mistresse Custance, the lorde you saue and kepe,
From me Roister Doister, whether I wake or slepe.
Who fauoureth you no lesse, (ye may be bolde)
Than this letter purporteth, which ye haue vnfolde.

Scene 5

Sweete mistresse, where as I loue you, nothing at all
Regarding your richesse and substance: chiefe of all
For your personage, beautie, demeanour and witte
I commende me vnto you: Neuer a whitte
Sory to heare reporte of your good welfare.
For (as I heare say) suche your conditions are,
That ye be worthie fauour: Of no liuing man
To be abhorred: of euery honest man
To be taken for a woman enclined to vice
Nothing at all: to vertue giuing hir due price.
Wherfore concerning mariage, ye are thought
Suche a fine Paragon, as nere honest man bought.
And nowe by these presents I doe you aduertise,
That I am minded to marrie you: In no wyse
For your goodes and substance: I can be content
To take you as you are: yf ye will be my wife,
Ye shall be assured for the time of my life,
I wyll keepe you right well: from good raiment and fare,
Ye shall not be kept: but in sorowe and care
Ye shall in no wyse lyue: at your owne libertie,
Doe and say what ye lust: ye shall neuer please me
But when ye are merrie: I will bee all sadde
When ye are sorie: I wyll be very gladde
When ye seeke your heartes ease: I will be vnkinde
At no time: in me shall ye muche gentlenesse finde.
But all things contrary to your will and minde
Shall be done otherwise: I wyll not be behynde
To speake: And as for all they that woulde do you wrong,
(I wyll so helpe and maintayne ye) shall not lyue long.
Nor any foolishe dolte shall cumber you, but I,
I, who ere say nay, wyll sticke by you tyll I die.
Thus good mistresse Custance, the lorde you saue and kepe.
From me Roister Doister, whether I wake or slepe,
Who fauoureth you no lesse, (ye may be bolde)
Than this letter purporteth, which ye haue vnfolde.

See also

Written by S

Tue, 2014-02-25 at 07:30:28

Posted in language, literature

Free, defiant, and without a security label

leave a comment »

James Mickens is a CS researcher (“Galactic Viceroy of Research Magnificence”) who among other things writes wrote for the online version of Usenix’s magazine ;login: (called ;login: logout, published every other month). Here are some of his articles:

  1. [May 2013] The Saddest Moment
  2. [July 2013] Mobile Computing Research Is a Hornet’s Nest of Deception and Chicanery
  3. [September 2013] The Slow Winter
  4. [November 2013] The Night Watch
  5. [January 2014] This World of Ours
  6. [March 2014] To Wash It All Away

Reading these is an epiphany akin to one’s first encounter with Dave Barry or Airplane!

(Plug by Raymond Chen too.)

See also this video, where he answers questions like “What’s the best piece of advice you have ever received?” (Answer: “The best piece of advice was probably ‘Stay out of jail’. That came from my dad.”)

Edit [2014-03-13]: Apparently the March 2014 column is his last for the magazine; updated post.

Written by S

Wed, 2014-01-08 at 13:53:11

The idea of logarithms, and the first appearance of e

with 2 comments

[Incomplete post: about 10% written, about 90% left.]

The notion of the number e, the exponential function e^x, and logarithms \log x are often conceptual stumbling blocks even to someone who has an otherwise solid understanding of middle-school mathematics.

Just what is the number e? How was it first calculated / where did it first turn up? Premature exposure to its numerical value

\displaystyle e \approx 2.718281828459045\dots

only serves to deepen the mysteriousness and to make it seem arbitrary.

Here a historical perspective helps: as is often the case, here too, the first appearance is simpler and more well-motivated than the accounts in dry textbooks. This is from this account by Matthew P. Wiener (originally posted on USENET somewhere, as quoted by MJD). I’m just going to quote it directly for now, and edit it later:

Napier, who invented logarithms, more or less worked out a table of logarithms to base \frac1e, as follows:

     0  1  2  3   4   5   6    7    8    9    10 ...
     1  2  4  8  16  32  64  128  256  512  1024 ...

The arithmetic progression in the first row is matched by a geometric progression in the second row. If, by any luck, you happen to wish to multiply 16 by 32, that just happen to be in the bottom row, you can look up their “logs” in the first row and add 4+5 to get 9 and then conclude 16·32=512.

For most practical purposes, this is useless. Napier realized that what one needs to multiply in general is 1+\epsilon for a base—the intermediate values will be much more extensive. For example, with base 1.01, we get:

       0 1.00   1 1.01   2 1.02   3 1.03   4 1.04   5 1.05
       6 1.06   7 1.07   8 1.08   9 1.09  10 1.10  11 1.12
      12 1.13  13 1.14  14 1.15  15 1.16  16 1.17  17 1.18
      18 1.20  19 1.21  20 1.22  21 1.23  22 1.24  23 1.26
      24 1.27  25 1.28  26 1.30  27 1.31  28 1.32  29 1.33
      30 1.35  31 1.36  32 1.37  33 1.39  34 1.40  35 1.42
      50 1.64  51 1.66  52 1.68  53 1.69  54 1.71  55 1.73
      94 2.55  95 2.57  96 2.60  97 2.63  98 2.65  99 2.68
     100 2.70 101 2.73 102 2.76 103 2.79 104 2.81 105 2.84

So if you need to multiply 1.27 by 1.33, say, just look up their logs, in this case, 24 and 29, add them, and get 53, so 1.27·1.33=1.69. For two/three digit arithmetic, the table only needs entries up to 9.99.

Note that e is almost there, as the antilogarithm of 100. The natural logarithm of a number can be read off from the above table, as just [approximately] \frac1{100} the corresponding exponent.

What Napier actually did was work with base .9999999. He spent 20 years computing powers of .9999999 by hand, producing a grand version of the above. That’s it. No deep understanding of anything, no calculus, and e pops up anyway—in Napier’s case, \frac1e was the 10 millionth entry. (To be pedantic, Napier did not actually use decimal points, that being a new fangled notion at the time.)

Later, in his historic meeting with Briggs, two changes were made. A switch to a base > 1 was made, so that logarithms would scale in the same direction as the numbers, and the spacing on the logarithm sides was chosen so that \log(10)=1. These two changes were, in effect, just division by -\log_e(10).

In other words, e made its first appearance rather implicitly.

(I had earlier read a book on Napier and come to the same information though a lot less clearly, here.)

I had started writing a series of posts leading up to an understanding of the exponential function e^x (here, here, here), but it seems to have got abandoned. Consider this one a contribution to that series.

Written by S

Wed, 2013-11-27 at 10:52:51

Posted in mathematics

Wodehouse on Conan Doyle

with 9 comments

I have noted before, while reading Right Ho, Jeeves, how much it draws from and parodies the Sherlock Holmes stories of Sir Arthur Conan Doyle. In fact, the whole book can be read as if Bertie Wooster is Sherlock Holmes, or at least that he imagines himself to be. Rereading it this way threw up a surprising number of examples (as did Psmith, Journalist), all the way from obvious ones like “You know my methods, Jeeves. Apply them.”, to references so subtle that it’s not clear whether Wodehouse is consciously parodying Sherlock Holmes, or it’s a simple case of one author influencing another. (But perhaps they only seem subtle to those of us who aren’t as steeped in the Holmesverse as the readers of the early 1900s would be.) And of course, when in his stories he directly mentions detectives (such as in The Man With Two Left Feet), it’s laid on thick:

He had never measured a footprint in his life, and what he did not know about bloodstains would have filled a library.


A detective is only human. The less of a detective, the more human he is. Henry was not much of a detective, and his human traits were consequently highly developed.

And I knew, too, (see previous post) that both authors enjoyed cricket, and even turned out occasionally for the same celebrity cricket team.

Despite all the preceding, I still was surprised by the existence of this:


An edition of The Sign of the Four, with introduction by P. G. Wodehouse!

It appears that PGW was a fan of ACD: in 1925, in a letter to his friend William Townend, he wrote:

“Conan Doyle, a few words on the subject of. Don’t you find as you age in the wood, as we are both doing, that the tragedy of your life is that your early heroes lose their glamour? As a lad in the twenties you worship old whoever-it-is, the successful author, and by the time you’re forty you find yourself blushing hotly at the thought that you could ever have admired the bilge he writes.
Now with Doyle I don’t have that feeling. I still revere hls work as much as ever. I used to think it swell, and I still think it swell.
And apart from his work, I admire Doyle so much as a man. I should call him definitely a great man, and I don’t imagine I’m the only one who thinks so.

And the introduction to The Sign of the Four was written in the 1970s, when Wodehouse must have been over 90. He echoes much the same lines. The full introduction is attached, pieced together from some rather excellent sources on the internet.

When I was starting out as a writer—this would be about the time Caxton invented the printing press—Conan Doyle was my hero. Others might revere Hardy and Meredith. I was a Doyle man, and I still am. Usually we tend to discard the idols of our youth as we grow older, but I have not had this experience with A.C.D. I thought him swell then, and I think him swell now.

We were great friends in those days, our friendship only interrupted when I went to live in America. He was an enthusiastic cricketer—he could have played for any first-class country—and he used to have cricket weeks at his place in the country, to which I was almost always invited. And after a day’s cricket and a big dinner he and I would discuss literature.

The odd thing was that though he could be expansive about his least known short stories–those in Round the Red Lamp, for instance—I could never get him to talk of Sherlock Holmes, and I think the legend that he disliked Sherlock must be true. It is with the feeling that he would not object that I have sometimes amused myself by throwing custard pies at that great man.

Recently I have taken up the matter of Holmes’s finances.

Let me go into the matter, in depth, as they say. I find myself arriving at a curious conclusion.

Have you ever considered the matter of Holmes’s financial affairs?

Here we have a man who evidently was obliged to watch the pennies, for when we are introduced to him he is, according to Doctor Watson’s friend Stamford, “bemoaning himself because he could not find someone to go halves in some nice rooms which he had found and which were too much for his purse.” Watson offers himself as a fellow lodger, and they settle down in—I quote—a couple of comfortable bedrooms and a large sitting room at 221B Baker Street.

Now I lived in similar rooms at the turn of the century, and I paid twenty-one shillings a week for bed, breakfast, and dinner. An extra bedroom no doubt made the thing come higher for Holmes and Watson, but thirty shillings must have covered the rent and vittles, and there was never any question of a man as honest as Watson failing to come up with his fifteen bob each Saturday. It follows, then, that allowing for expenditures in the way of Persian slippers, tobacco, disguises, revolver cartridges, cocaine, and spare violin strings Holmes would have been getting by on a couple of pounds or so weekly. And with this modest state of life he appeared to be perfectly content. Let us take a few instances at random and see what he made as a “consulting detective.”

In the very early days of their association, using it as his “place of business,” he interviewed in the sitting room “a grey-headed seedy visitor, who was followed by a slipshod elderly woman, and after that a railway porter in his velveteen uniform.” Not much cash in that lot, and things did not noticably improve later, for we find his services engaged by a stenographer, a city clerk, a Greek interpreter, a landlady, and a Cambridge undergraduate.

So far from making money as a consulting detective, he must have been a good deal out of pocket most of the time. In A Study in Scarlet, Inspector Gregson asks him to come to 3 Lauriston Gardens in the Brixton neighborhood, because there has been “a bad business” there during the night. Off goes Holmes in a hansom cab from Baker Street to Brixton, a fare of several shillings, dispatches a long telegram (another two or three bob to the bad), summons “half a dozen of the dirtiest and most ragged street Arabs I ever clapped eyes on,” gives each of them a shilling, and tips a policeman half a sovereign. The whole affair must have cost him considerably more than a week’s rent at Baker Street, and no hope of getting any of it back from Inspector Gregson, for Gregson, according to Holmes himself, was “one of the smartest of all the Scotland Yarders.”

Inspector Gregson! Inspector Lestrade! Those clients! I found myself thinking a good deal about them, and it was not long before the truth dawned upon me, that they were merely cheap actors, hired to deceive doctor Watson, who had to be deceived because he had the job of writing the stories.

For what would the ordinary private investigator have said to himself when starting out in business? He would have said ‘Before I take on work for a client I must be sure that the client has the stuff. The daily sweetener and the little something down in advance are of the essence,’ and he would have had those landladies and those Greek interpreters out of his sitting room before you could say ‘bloodstain.’ Yet Holmes, who could not afford a pound a week for lodgings, never bothered. Significant!

Later the thing became absolutely farcical, for all pretence that he was engaged in a gainful occupation was dropped by himself and the clients. I quote Doctor Watson.

“He tossed a crumpled letter across the table to me. It was dated from Montague Place upon the preceding evening and ran thus:

Dear Mr. Holmes,
I am anxious to consult you as to whether or not I should accept a situation which has been offered to me as a governess.
I shall call at half-past ten tomorrow, if I do not inconvenience you.
Yours faithfully
Violet Hunter.”

Now, the fee an investigator could expect from a governess, even one in full employment, could scarcely be more than a few shillings, yet when two weeks later Miss Hunter wired “Please be at the Black Swan at Winchester at mid-day tomorrow,” Holmes dropped everything and sprang into the 9:30 train.

It all boils down to one question–Why is a man casual about money?

The answer is–Because he has a lot of it.

Had Holmes?

He pretended he hadn’t, but that was merely the illusion he was trying to create because he needed a front for his true activities. He was pulling the stuff in from another source. Where is the big money? Where it has always been, in crime. Bags of it, and no income tax. If you want to salt away a few million for a rainy day, you don’t spring into 9:30 trains to go and talk to governesses, you become a Master Criminal, sitting like a spider in the center of its web and egging your corps of assistants on to steal jewels and navel treaties. I saw daylight, and all the pieces of the jigsaw puzzle fell into place. Holmes was Professor Moriarty.

What was that name again?

Professor Moriarty.

Do you mean that man who was forever oscillating his face from side to side in a curiously reptilian fashion?

That’s the one.

But Holmes’ face didn’t forever oscillate from side to side in a curiously reptilian fashion.

Nor did Professor Moriarty’s.

Holmes said it did.

And to whom? To Doctor Watson, in order to ensure that the misleading description got publicity. Watson never saw Moriarty. All he knew about him was what Holmes told him on the evening of April 24,1891. And Holmes made a little slip on the occasion. He said that on his way to see Watson he had been attacked by a rough with a bludgeon. A face-oscillating napoleon of Crime, anxious to eliminate someone he disliked, would have thought up something better than roughs with bludgeons. Dropping cobras down the chimney is the mildest thing that would have occurred to him.

P.S. Just kidding, boys. Actually, like all the rest of you, I am never happier than when curled up with Sherlock Holmes, and I hope Messrs Ballantine will sell several million of him. As the fellow said, there’s no police like Holmes.

–P.G. Wodehouse.


Click to access v7n2Seven.pdf

Other stuff:

(All written during the time between the publication of Sherlock Holme’s death in FINA published December 1893, and his reapparance in EMPT published September 1903. The Hound of the Baskervilles had been serialized from August 1901 to April 1902. Doyle had announced the impending return of Sherlock Holmes in the Strand, whcih is why Wodehouse wrote “Back to his Native Strand”.)

* Wodehouse wrote (unsigned) a parody called “Dudley Jones, Bore Hunter” (, in Punch on April 29, 1903 ( and May 6, 1903 (

* Wodehouse wrote (unsigned) a poem called “Back to his Native Strand” for Punch on May 27, 1903:

* Wodehouse did an “interview” of ACD in “VC” magazine, July 2, 1903:

* Wodehouse wrote (unsigned) “The Prodigal” for Punch on September 23, 1903:

Written by S

Fri, 2013-10-11 at 08:37:03

Posted in literature

Cricket poems

with 3 comments

Arthur Conan Doyle played 10 first-class matches between 1900 (when he was over 40) and 1907, playing for the MCC. He averaged close to 20 with the bat, with a high score of 43. On 25 August 1900, against London County at Crystal Palace, he took his only first-class wicket: that of W. G. Grace, who was batting on 110 at the time (and declared his team’s innings immediately after getting out). He wrote a poem about it.

A Reminiscence of Cricket

Once in my heyday of cricket,
One day I shall ever recall!
I captured that glorious wicket,
The greatest, the grandest of all.

Before me he stands like a vision,
Bearded and burly and brown,
A smile of good humoured derision
As he waits for the first to come down.

A statue from Thebes or from Knossos,
A Hercules shrouded in white,
Assyrian bull-like colossus,
He stands in his might.

With the beard of a Goth or a Vandal,
His bat hanging ready and free,
His great hairy hands on the handle,
And his menacing eyes upon me.

And I – I had tricks for the rabbits,
The feeble of mind or eye,
I could see all the duffer’s bad habits
And where his ruin might lie.

The capture of such might elate one,
But it seemed like one horrible jest
That I should serve tosh to the great one,
Who had broken the hearts of the best.

Well, here goes! Good Lord, what a rotter!
Such a sitter as never was dreamt;
It was clay in the hands of the potter,
But he tapped it with quiet contempt.

The second was better – a leetle;
It was low, but was nearly long-hop;
As the housemaid comes down on the beetle
So down came the bat with a chop.

He was sizing me up with some wonder,
My broken-kneed action and ways;
I could see the grim menace from under
The striped peak that shaded his gaze.

The third was a gift or it looked it—
A foot off the wicket or so;
His huge figure swooped as he hooked it,
His great body swung to the blow.

Still when my dreams are night-marish,
I picture that terrible smite,
It was meant for a neighboring parish,
Or any place out of sight.

But – yes, there’s a but to the story –
The blade swished a trifle too low;
Oh wonder, and vision of glory!
It was up like a shaft from a bow.

Up, up like a towering game bird,
Up, up to a speck in the blue,
And then coming down like the same bird,
Dead straight on the line that it flew.

Good Lord, it was mine! Such a soarer
Would call for a safe pair of hands;
None safer than Derbyshire Storer,
And there, face uplifted, he stands

Wicket keep Storer, the knowing,
Wary and steady of nerve,
Watching it falling and growing
Marking the pace and curve.

I stood with my two eyes fixed on it,
Paralysed, helpless, inert;
There was ‘plunk’ as the gloves shut upon it,
And he cuddled it up to his shirt.

Out – beyond question or wrangle!
Homeward he lurched to his lunch!
His bat was tucked up at an angle,
His great shoulders curved to a hunch.

Walking he rumbled and grumbled,
Scolding himself and not me;
One glove was off, and he fumbled,
Twisting the other hand free

Did I give Storer the credit
The thanks he so splendidly earned?
It was mere empty talk if I said it,
For Grace had already returned.

Incidentally, W. G., like Conan Doyle, was also a doctor with no time for that profession. Here’s another article about Conan Doyle. He also made up a story about a “high dropping full toss” (lob bowling?) that fell on the stumps from the air. (Discussion.)

P. G. Wodehouse wrote a happy little poem about a fielder who misses a catch.


The sun in the heavens was beaming,
The breeze bore an odour of hay,
My flannels were spotless and gleaming,
My heart was unclouded and gay;
The ladies, all gaily apparelled,
Sat round looking on at the match,
In the tree-tops the dicky-birds carolled,
All was peace — till I bungled that catch.

My attention the magic of summer
Had lured from the game — which was wrong.
The bee (that inveterate hummer)
Was droning its favourite song.
I was tenderly dreaming of Clara
(On her not a girl is a patch),
When, ah, horror! there soared through the air a
Decidedly possible catch.

I heard in a stupor the bowler
Emit a self-satisfied ‘Ah!’
The small boys who sat on the roller
Set up an expectant ‘Hurrah!’
The batsman with grief from the wicket
Himself had begun to detach —
And I uttered a groan and turned sick. It
Was over. I’d buttered the catch.

O, ne’er, if I live to a million,
Shall I feel such a terrible pang.
From the seats on the far-off pavilion
A loud yell of ecstasy rang.
By the handful my hair (which is auburn)
I tore with a wrench from my thatch,
And my heart was seared deep with a raw burn
At the thought that I’d foozled that catch.

Ah, the bowler’s low, querulous mutter
Points loud, unforgettable scoff!
Oh, give me my driver and putter!
Henceforward my game shall be golf.
If I’m asked to play cricket hereafter,
I am wholly determined to scratch.
Life’s void of all pleasure and laughter;
I bungled the easiest catch.

Both Conan Doyle and Wodehouse played cricket at one point for J. M. Barrie’s team Allah-akbarries (named in the belief that “Allahu Akbar” meant “God help us!”, but of course probably more for the “barries” in the name), some of whose other players included Rudyard Kipling, H. G. Wells, G. K. Chesterton, Jerome K. Jerome, A. A. Milne.

Here’s another great article about Conan Doyle and Wodehouse.

A. A. Milne wrote some poems about cricket as well.

An article about authors.

Casey at the Bat is the most famous baseball poem. (Wikipedia article)

Written by S

Sun, 2013-09-01 at 14:35:50

Posted in literature

Tagged with

Stephen Fry’s “The Ode Less Travelled”: Foreword

with one comment

I reproduce below the line below Stephen Fry’s entire foreword to his book The Ode Less Travelled, because I find myself frequently referring to it and would like to be able to direct friends to some place to read it‌ — this is now such a place.

I HAVE A DARK AND DREADFUL SECRET. I write poetry. This is an embarrassing confession for an adult to make. In their idle hours Winston Churchill and Noël Coward‎ painted. For fun and relaxation Albert Einstein played the violin. Hemingway hunted, Agatha Christie gardened, James Joyce sang arias and Nabokov chased butterflies. But poetry?

I have a friend who drums in the attic, another who has been building a boat for years. An actor I know is prouder of the reproduction eighteenth-century duelling pistols he makes in a small workshop than he is of his knighthood. Britain is a nation of hobbyists—eccentric amateurs, talented part-timers, Pooterish potterers and dedicated autodidacts in every field of human endeavour. But poetry?

An adolescent girl may write poetry, so long as it is securely locked up in her pink leatherette five-year diary. Suburban professionals are permitted to enter jolly pastiche competitions in the Spectator and New Statesman. At a pinch, a young man may be allowed to write a verse or two of dirty doggerel and leave it on a post-it note stuck to the fridge when he has forgotten to buy a Valentine card. But that’s it. Any more forays into the world of Poesy and you release the beast that lurks within every British breast—and the name of the beast is Embarrassment.

And yet…

I believe poetry is a primal impulse within us all. I believe we are all capable of it and furthermore that a small, often ignored corner of us positively yearns to try it. I believe our poetic impulse is blocked by the false belief that poetry might on the one hand be academic and technical and on the other formless and random. It seems to many that while there is a clear road to learning music, gardening or watercolours, poetry lies in inaccessible marshland: no pathways, no signposts, just the skeletons of long-dead poets poking through the bog and the unedifying sight of living ones floundering about in apparent confusion and mutual enmity. Behind it all, the dread memory of classrooms swollen into resentful silence while the English teacher invites us to ‘respond’ to a poem.

For me the private act of writing poetry is songwriting, confessional, diary-keeping, speculation, problem-solving, storytelling, therapy, anger management, craftsmanship, relaxation, concentration and spiritual adventure all in one inexpensive package.

Suppose I want to paint but seem to have no obvious talent. Never mind: there are artist supply shops selling paints, papers, pastels, charcoals and crayons. There are ‘How To’ books everywhere. Simple lessons in the rules of proportion and guides to composition and colourmixing can make up for my lack of natural ability and provide painless technical grounding. I am helped by grids and outlines, pantographs and tracing paper; precise instructions guide me in how to prepare a canvas, prime it with paint and wash it into an instant watercolour sky. There are instructional videos available; I can even find channels on cable and satellite television showing gentle hippies painting lakes, carving pine trees with palette knives and dotting them with impasto snow. Mahlsticks, sable, hogs-hair, turpentine and linseed. Viridian, umber, ochre and carmine. Perspective, chiaroscuro, sfumato, grisaille, tondo and morbidezza. Reserved modes and materials. The tools of the trade. A new jargon to learn. A whole initiation into technique, form and style.

Suppose I want to play music but seem to have no obvious talent. Never mind: there are music shops selling instruments, tuning forks, metronomes and ‘How To’ books by the score. And scores by the score. Instructional videos abound. I can buy digital keyboards linked to programmes that plug into my computer and guide me through the rudiments, monitoring my progress and accuracy. I start with scales and move on to chords and arpeggios. There are horsehair, rosin and catgut, reeds, plectrums and mouthpieces. There are diminished sevenths, augmented fifths, relative minors, trills and accidentals. There are riffs and figures, licks and vamps. Sonata, adagio, crescendo, scherzo and twelve-bar blues. Reserved modes and materials. The tools of the trade. A new jargon to learn. A whole initiation into technique, form and style.

To help us further there are evening classes, clubs and groups. Pack up your easel and palette and go into the countryside with a party of like-minded enthusiasts. Sit down with a friend and learn a new chord on the guitar. Join a band. Turn your watercolour view of Lake Windermere into a tablemat or T-shirt. Burn your version of ‘Stairway to Heaven’ onto a CD and alarm your friends.

None of these adventures into technique and proficiency will necessarily turn you into a genius or even a proficient craftsman. Your view of Snow on York Minster, whether languishing in the loft or forming the basis of this year’s Christmas card doesn’t make you Turner, Constable or Monet. Your version of ‘Fur Elise’ on electric piano might not threaten Alfred Brendel, your trumpet blast of ‘Basin Street Blues’ could be so far from Satchmo that it hurts and your take on ‘Lela’ may well stand as an eternal reproach to all those with ears to hear. You may not sell a single picture, be invited even once to deputise for the church organist when she goes down with shingles or have any luck at all when you try out for the local Bay City Rollers tribute band. You are neither Great Artist, sessions professional, illustrator or admired amateur.

So what? You are someone who paints a bit, scratches around on the keyboard for fun, gets a kick out of learning a tune or discovering a new way of rendering the face of your beloved in charcoal. You have another life, you have family, work and friends but this is a hobby, a pastime, FUN. Do you give up the Sunday kick-around because you’ll never be Thierry Henry? Of course not. That would be pathologically vain. We don’t stop talking about how the world might be better just because we have no chance of making it to Prime Minister. We are all politicians. We are all artists. In an open society everything the mind and hands can achieve is our birthright. It is up to us to claim it.

And you know, you might be the real thing, or someone with the potential to give as much pleasure to others as you derive yourself. But how you will ever know if you don’t try?

As the above is true of painting and music, so it is true of cookery and photography and gardening and interior decoration and chess and poker and skiing and sailing and carpentry and bridge and wine and knitting and brass-rubbing and line-dancing and the hundreds of other activities that enrich and enliven the daily toil of getting and spending, mortgages and shopping, school and office. There are rules, conventions, techniques, reserved objects, equipment and paraphernalia, time-honoured modes, forms, jargon and tradition. The average practitioner doesn’t expect to win prizes, earn a fortune, become famous or acquire absolute mastery in their art, craft, sport-or as we would say now, their chosen leisure pursuit. It really is enough to have fun.

The point remains: it isn’t a burden to learn the difference between acid and alkaline soil or understand how f-stops and exposure times affect your photograph. There’s no drudgery or humiliation in discovering how to knit, purl and cast off, snowplough your skis, deglaze a pan, carve a dovetail or tot up your bridge hand according to Acol. Only an embarrassed adolescent or deranged coward thinks jargon and reserved languages are pretentious and that detail and structure are boring. Sensible people are above simpering at references to colour in music, structure in wine or rhythm in architecture. When you learn to sail you are literally shown the ropes and taught that they are called sheets or painters and that knots are hitches and forward is aft and right is starboard. That is not pseudery or exclusivity, it is precision, it is part of initiating the newcomer into the guild. Learning the lingo is the beginning of our rite of passage.

In music, tempo is not the same as rhythm, which is not the same as pulse. There are metronomic indications and time signatures. At some point along the road between picking out a tune with one finger and really playing we need to know these distinctions. For some it comes naturally and seems inborn, for most of us the music is buried deep inside but needs a little coaxing and tuition to be got out. So someone shows us, or we progress by video, evening class or book. Talent is inborn but technique is learned.

Talent without technique is like an engine without a steering wheel, gears or brakes. It doesn’t matter how thoroughbred and powerful the V12 under the bonnet if it can’t be steered and kept under control. Talented people who do nothing with their gifts often crash and burn. A great truth, so obvious that it is almost a secret, is that most people are embarrassed to the point of shame by their talents. Ashamed of their gifts but proud to bursting of their achievements. Do athletes boast of their hand-eye coordination, grace and natural sense of balance? No, they talk of how hard they trained, the sacrifices they made, the effort they put in.

Ah, but a man’s reach should exceed his grasp
Or what’s a heaven for?

Robert Browning’s cry brings us back, at last, to poetry. While it is perfectly possible that you did not learn music at school, or drawing and painting, it is almost certain that you did learn poetry. Not how to do it, almost never how to write your own, but how, God help us, to appreciate it.

We have all of us, all of us, sat with brows furrowed feeling incredibly dense and dumb as the teacher asks us to respond to an image or line of verse.

What do you think Wordsworth was referring to here?
What does Wilfred Owen achieve by choosing this metaphor?
How does Keats respond to the nightingale?
Why do you think Shakespeare uses the word ‘gentle’ as a verb?
What is Larkin’s attitude to the hotel room?

It brings it all back, doesn’t it? All the red-faced, blood-pounding humiliation and embarrassment of being singled out for comment.

The way poetry was taught at school reminded W. H. Auden of a Punch cartoon composed, legend has it, by the poet A. E. Housman. Two English teachers are walking in the woods in springtime. The first, on hearing birdsong, is moved to quote William Wordsworth:

TEACHER 1: Oh cuckoo, shall I call thee bird
Or but a wandering voice?
TEACHER 2: State the alternative preferred
With reasons for your choice.

Even if some secret part of you might have been privately moved and engaged, you probably went through a stage of loathing those bores Shakespeare, Keats, Owen, Eliot, Larkin and all who came before and after them. You may love them now, you may still hate them or perhaps you feel entirely indifferent to the whole pack of them. But however well or badly we were taught English literature, how many of us have ever been shown how to write our own poems?

Don’t worry, it doesn’t have to rhyme. Don’t bother with metre and verses. Just express yourself. Pour out your feelings.

Suppose you had never played the piano in your life.

Don’t worry, just lift the lid and express yourself. Pour out your feelings.

We have all heard children do just that and we have all wanted to treat them with great violence as a result. Yet this is the only instruction we are ever likely to get in the art of writing poetry: Anything goes.

But that’s how modern poetry works, isn’t it? Free verse, don’t they call it? Vers libre?

Ye-e-es…And in avant-garde music, John Cage famously wrote a piece of silence called ‘4 Minutes 33 Seconds’ and created other works requiring ball-bearings and chains to be dropped on to prepared pianos. Do music teachers suggest that to children? Do we encourage them to ignore all harmony and rhythm and just make noise? It is important to realise that Cage’s first pieces were written in the Western compositional tradition, in movements with conventional Italian names like lento, vivace and fugato. Picasso’s early paintings are flawless models of figurative accuracy. Listening to music may inspire an extraordinary emotional response, but extraordinary emotions are not enough to make music.

Unlike musical notation, paint or clay, language is inside every one of us. For free. We are all proficient at it. We already have the palette, the paints and the instruments. We don’t have to go and buy any reserved materials. Poetry is made of the same stuff you are reading now, the same stuff you use to order pizza over the phone, the same stuff you yell at your parents and children, whisper in your lover’s ear and shove into an e-mail, text or birthday card. It is common to us all. Is that why we resent being told that there is a technique to its highest expression, poetry? I cannot ski, so I would like to be shown how to. I cannot paint, so I would value some lessons. But I can speak and write, so do not waste my time telling me that I need lessons in poetry, which is, after all, no more than emotional writing, with or without the odd rhyme. Isn’t it?

Jan Schreiber in a review of Timothy Steele’s Missing Measures, says this of modern verse:

The writing of poetry has been made laughably easy. There are no technical constraints. Knowledge of the tradition is not necessary, nor is a desire to communicate, this having been supplanted in many practitioners by the more urgent desire to express themselves. Even sophistication in the manipulation of syntax is not sought. Poetry, it seems, need no longer be at least as well written as prose.

Personally, I find writing without form, metre or rhyme not ‘laughably easy’ but fantastically difficult. If you can do it, good luck to you and farewell, this book is not for you: but a word of warning from W.H. Auden before you go.

The poet who writes ‘free’ verse is like Robinson Crusoe on his desert island: he must do all his cooking, laundry and darning for himself. In a few exceptional cases, this manly independence produces something original and impressive, but more often the result is squalor—dirty sheets on the unmade bed and empty bottles on the unswept floor.

I cannot teach you how to be a great poet or even a good one. Dammit, I can’t teach myself that. But I can show you how to have fun with the modes and forms of poetry as they have developed over the years. By the time you have read this book you will be able to write a Petrarchan sonnet, a Sapphic Ode, a ballade, a villanelle and a Spenserian stanza, among many other weird and delightful forms; you will be confident with metre, rhyme and much else besides. Whether you choose to write on the stupidity of advertising, the curve of your true love’s buttocks, the folly of war or the irritation of not being able to open a pickle jar is unimportant. I will give you the tools, you can finish the job. And once you have got the hang of the forms, you can devise your own. The Robertsonian Sonnet. The Jonesian Ode. The Millerian Stanza.

This is not an academic book. It is unlikely to become part of the core curriculum. It may help you with your English exams because it will certainly allow you to be a smart-arse in Practical Criticism papers (if such things still exist) and demonstrate that you know a trochee from a dactyl, a terza from an ottava rima and assonance from enjambment, in which case I am happy to be of service. It is over a quarter of a century since I did any teaching and I have no idea if such knowledge is considered good or useless these days, for all I know it will count against you.

I have written this book because over the past thirty-five years I have derived enormous private pleasure from writing poetry and like anyone with a passion I am keen to share it. You will be relieved to hear that I will not be burdening you with any of my actual poems (except sample verse specifically designed to help clarify form and metre): I do not write poetry for publication, I write it for the same reason that, according to Wilde, one should write a diary, to have something sensational to read on the train. And as a way of speaking to myself. But most importantly of all for pleasure.

This is not the only work on prosody (the art of versification) ever published in English, but it is the one that I should like to have been available to me many years ago. It is technical, yes, inasmuch as it investigates technique, but I hope that does not make it dry, obscure or difficult-after all, ‘technique’ is just the Greek for ‘art’. I have tried to make everything approachable without being loopily matey or absurdly simplistic.

I certainly do not attempt in this book to pick up where those poor teachers left off and instruct you in poetry appreciation. I suspect, however, that once you have started writing a poem of any real shape you will find yourself admiring and appreciating other poets’ work a great deal more. If you have never picked up a golf club you will never really know just how remarkable Ernie Els is (substitute tennis racket for Roger Federer, frying pan for Gordon Ramsay, piano for Jools Holland and so on).

But maybe you are too old a dog to learn new tricks? Maybe you have missed the bus? That’s hooey. Thomas Hardy (a finer poet than he was a novelist in my view) did not start publishing verse till he was nearly sixty.

Every child is musical. Unfortunately this natural gift is squelched before it has time to develop. From all my life experience I remember being laughed at because my voice and the words I sang didn’t please someone. My second grade teacher, Miss Stone would not let me sing with the rest of the class because she judged my voice as not musical and she said I threw the class off key. I believed her which led to the blockage of my appreciation of music and blocked my ability to write poetry. Fortunately at the age of 57 I had a significant emotional event which unblocked my ability to compose poetry which many people believe has lyrical qualities.

So writes one Sidney Madwed. Mr Madwed may not be Thomas Campion or Cole Porter, but he believes that an understanding of prosody has set him free and now clearly has a whale of a time writing his lyrics and verses. I hope reading this book will take the place for you of a ‘significant emotional event’ and awaken the poet that has always lain dormant within.

It is never too late. We are all opsimaths.

Opsimath, noun: one who learns late in life.

Let us go forward together now, both opsimathically and optimistically. Nothing can hold us back. The ode beckons.

Written by S

Sun, 2013-08-04 at 18:27:35

Posted in language, literature, quotes

The functional equation f(x+y) = f(x)f(y)

with 3 comments

Suppose f: \mathbb{R} \to \mathbb{R} satisfies f(x+y) = f(x) f(y). What can we say about f?

Putting y = 0 gives

\displaystyle f(x) = f(x+0) = f(x)f(0),

which can happen if either f(x) = 0 or f(0) = 1. Note that the function f which is identically zero satisfies the functional equation. If f is not this function, i.e., if f(x) \neq 0 for at least one value of x, then plugging that value of x (say x^*) into the equation gives f(0) = 1. Also, for any x, the equation f(x^*) = f(x +x^* - x) = f(x)f(x^* - x) forces f(x) \neq 0 as well. Further, f(x) = f(x/2 + x/2) = f(x/2)^2 so f(x) > 0 for all x.

Next, putting y = x gives f(2x) = f(x)^2, and by induction f(nx) = f(x)^n. Putting \frac{x}{n} in place of x in this gives f(n\frac{x}{n}) = f(\frac{x}{n})^n which means f(\frac{x}{n}) = f(x)^{\frac1n} (note we’re using f(x) > 0 here). And again, f(\frac{m}{n}x) = f(x)^{m/n}. So f(\frac{m}{n}) = f(1)^{m/n}, which completely defines the function at rational points.

[As f(1) > 0, it can be written as f(1) = e^k for some constant k, which gives f(x) = e^{kx} for rational x.]

To extend this function to irrational numbers, we need some further assumptions on f, such as continuity. It turns out that being continuous at any point is enough (and implies the function is f(x) = f(1)^x everywhere): note that f(x + m/n) = f(x)f(m/n) = f(x)f(1)^{m/n}. Even being Lebesgue-integrable/measurable will do.

Else, there are discontinuous functions satisfying the functional equation. (Basically, we can define the value of the function separately on each “independent” part. That is, define the equivalence class where x and y are related if y = r_1x + r_2 for rationals r_1 and r_2, pick a representative for each class using the axiom of choice (this is something like picking a basis for \mathbb{R}/\mathbb{Q}, which corresponds to the equivalence class defined by the relation y = r_1x), define the value of the function independently for each representative, and this fixes the value of f on \mathbb{R}. See this article for more details.)

To step back a bit: what the functional equation says is that f is a homorphism from (\mathbb{R}, +), the additive group of real numbers, to (\mathbb{R}, \times), the multiplicative monoid of real numbers. If f is not the trivial identically-zero function, then (as we saw above) f is in fact a homomorphism from (\mathbb{R}, +), the additive group of real numbers, to (\mathbb{R_+^*}, \times), the multiplicative group of positive real numbers. What we proved is that the exponential functions e^{kx} are precisely all such functions that are nice (nice here meaning either measurable or continuous at least one point). (Note that this set includes the trivial homomorphism corresponding to k = 0: the function f(x) = 1 identically everywhere. If f is not this trivial map, then it is in fact an isomorphism.)

Edit [2013-10-11]: See also Overview of basic facts about Cauchy functional equation.

Written by S

Mon, 2013-04-08 at 11:24:08

Posted in mathematics

Trajectory of a point moving with acceleration perpendicular to velocity

with 8 comments

(Just some basic high-school physics stuff; to assure myself I can still do some elementary things. :P Essentially, showing that if a particle moves with acceleration perpendicular to velocity, or velocity perpendicular to position, then it traces out a circle. Stop reading here if this is obvious.)

Suppose a point moves in the plane such that its acceleration is always perpendicular to its velocity, and of the same magnitude. What is its path like?

To set up notation: let’s say the point’s position at time t is (p_x(t), p_y(t)), its velocity is (v_x(t), v_y(t)) = \left(\frac{d}{dt}p_x(t), \frac{d}{dt}p_y(t)\right), and its acceleration is (a_x(t), a_y(t)) = \left(\frac{d}{dt}v_x(t), \frac{d}{dt}v_y(t)\right).

The result of rotating a point (x,y) by 90° is (-y, x). (E.g. see figure below)


So the fact that acceleration is at right angles to velocity means that (a_x(t), a_y(t)) = (-v_y(t), v_x(t)), or, to write everything in terms of the velocity,

\begin{aligned} \frac{d}{dt}v_x(t) &= -v_y(t) \\  \frac{d}{dt}v_y(t) &= v_x(t) \end{aligned}

where we can get rid of v_x(t) by substituting the second equation (in the form v_x(t) = \frac{d}{dt}v_y(t)) into the first:

v_y(t) = -\frac{d}{dt}v_x(t) = -\frac{d}{dt}\left(\frac{d}{dt}v_y(t)\right)

or in other words

v_y(t) = -\frac{d^2}{dt^2}v_y(t).

By some theory about ordinary differential equations, which I don’t know (please help!) (but see the very related example you saw in high school, of simple harmonic motion), the solutions to this equation are \sin(t) and \cos(t) and any linear combination of those: the solution in general is

\begin{aligned}  v_y(t) &= a \sin(t) + b \cos(t) \\  &= \sqrt{a^2 + b^2} \left(\frac{a}{\sqrt{a^2+b^2}}\sin(t) + \frac{b}{\sqrt{a^2+b^2}}\cos(t)\right) \\  &= R\sin (t + \alpha)  \end{aligned}

where R = \sqrt{a^2 + b^2} and \alpha is the angle such that \cos(\alpha) = \frac{a}{\sqrt{a^2+b^2}} and \sin(\alpha) = \frac{b}{\sqrt{a^2+b^2}}. And the fact that v_x(t) = \frac{d}{dt}v_y(t) gives v_x(t) = R\cos(t + \alpha). So (v_x(t), v_y(t)) = (R\cos(t + \alpha), R\sin(t + \alpha)). Note that (a_x(t), a_y(t)) = \left(\frac{d}{dt}v_x(t), \frac{d}{dt}v_y(t)\right) = (-R\sin(t+\alpha), R\cos(t+\alpha)) is indeed perpendicular to (v_x(t), v_y(t)) as we wanted.

The actual trajectory (p_x(t), p_y(t)) can be got by integrating

\left(\frac{d}{dt}p_x(t), \frac{d}{dt}p_y(t)\right)  = (v_x(t), v_y(t)) = (R\cos(t + \alpha), R\sin(t + \alpha))

to get p_x(t) = R\sin(t + \alpha) + c_1 and p_y(t) = -R\cos(t + \alpha) + c_2. This trajectory is a point moving on a circle centered at point (c_1, c_2) and of radius R, with speed R or unit angular speed. Note that velocity is also perpendicular to the point’s position wrt the centre of the circle, as velocity is tangential to the circle, as it should be.

With a suitable change of coordinates (translate the origin to (c_1, c_2), then rotate the axes by \frac{\pi}{2}+\alpha, then scale everything so that R = 1), this is the familiar paremetrization (\cos(t), \sin(t)) of the circle.

Note: Just as we derived (v_x(t), v_y(t)) = (R\cos(t + \alpha), R\sin(t + \alpha)) from assuming that the acceleration is perpendicular to velocity, we can, by assuming that velocity is perpendicular to position, identically derive (p_x(t), p_y(t)) = (R\cos(t + \alpha), R\sin(t + \alpha)), i.e. that the point moves on a circle.

Written by S

Sun, 2013-04-07 at 23:38:01

Posted in mathematics

Typing Kannada on Mac OS X

with 26 comments

(Thanks to this and this.)

Turns out it’s very easy, and we can basically use the same input method (UIM) as in Linux.

  1. Get MacUIM from its website
  2. Install it.
  3. Go to System Preferences -> Language & Text -> Input Sources System Preferences -> Keyboard -> Input Sources, and turn on MacUIM. (Click + and add “MacUIM (Roman)” under English.) Tick “Show Input menu in menu bar” too. Screen Shot 2013-04-07 at 12.33.19 AM
  4. I now have three input methods: US, EasyIAST (see earlier post), and MacUIM (Roman).
  5. Go to System Preferences -> MacUIM -> General, and in Input method, choose m17n-kn-itrans
  6. Go to System Preferences -> MacUIM -> Helper, tick “Use Helper-Applet”, and in the list at the right, tick m17n-kn-itrans.
    Screen Shot 2013-04-07 at 5.45.57 PM
  7. [Just for me] I have some changes to kn-itrans.mim, to make it closer to HK (and remove nonsense like “RRi” or whatnot just to type ಋ): download this file kn-itrans.mim, and remove the pdf extension. It goes into /Library/M17NLib/share/m17n/kn-itrans.mim

Written by S

Sun, 2013-04-07 at 01:14:35

Posted in compknow

Plastic wire baskets

with 5 comments

These used to be ubiquitous a while ago (IIRC, I used to carry one of these daily to school as my lunch basket at some point; we still have one such basket at home), but photos seem hard to find on the internet (or I’m just missing the right keywords). So, photos:

Poorani Ammal, Copyright  R Revathi / Mylapore Times

Poorani Ammal, Copyright R Revathi / Mylapore Times

Written by S

Sun, 2013-04-07 at 00:14:29

Posted in Uncategorized

The power series for sin and cos

with one comment

There are many ways to derive the power series of \sin x and \cos x using the machinery of Taylor series etc., but below is another elementary way of demonstrating that the well-known power series expansions are the right ones. The argument below is from Tristan Needham’s Visual Complex Analysis, which I’m reproducing without looking at the book just to convince myself that I’ve internalized it correctly.

So: let
\displaystyle  \begin{aligned}   P(x) &= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots \quad \text{ and }\\  Q(x) &= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots .  \end{aligned}

We will take the following two for granted (both can be proved with some effort):

  1. Both power series are convergent.
  2. The power series can be differentiated term-wise.

As suggested by (2) above, the first thing we observe is that \frac{d}{dx}P(x) = Q(x) and \frac{d}{dx}Q(x) = -P(x).

So firstly:
\begin{aligned}  \frac{d}{dx}(P(x)^2 + Q(x)^2)   &= 2P(x)P'(x) + 2Q(x)Q'(x) \\  &= 2P(x)Q(x) - 2Q(x)P(x) \\  &= 0  \end{aligned}
which means that P(x)^2 + Q(x)^2 is a constant and does not vary with x. Putting x = 0 shows that P(0) = 0 and Q(0) = 1, so P(x)^2 + Q(x)^2 = 1 for all x.

Secondly, define the angle \theta as a function of x, by \tan \theta(x) = P(x)/Q(x). (To be precise, this defines \theta(x) up to a multiple of \pi, i.e. modulo \pi.)
Differentiating the left-hand side of this definition gives
\displaystyle \begin{aligned}  \frac{d}{dx} \tan \theta(x)   &= (1 + \tan^2 \theta(x)) \theta'(x) \\  &= (1 + \frac{P(x)^2}{Q(x)^2}) \theta'(x) \\  &= \frac{1}{Q(x)^2} \theta'(x)   \end{aligned}
(where \theta'(x) means \frac{d}{dx} \theta(x))
while differentiating the right-hand side gives
\displaystyle \begin{aligned}  \frac{d}{dx} \frac{P(x)}{Q(x)}   &= \frac{Q(x)P'(x) - P(x)Q'(x)}{Q(x)^2} \\  &= \frac{Q(x)Q(x) + P(x)P(x)}{Q(x)^2} \\  &= \frac{1}{Q(x)^2}  \end{aligned}

The necessary equality of the two tells us that \frac{d}{dx}\theta(x) = 1, which along with the initial condition \tan \theta(0) = P(0)/Q(0) = 0 = \tan 0 that says \theta(0) \equiv 0 \mod \pi, gives \theta(x) = x (or to be precise, \theta(x) \equiv x \pmod {\pi}).

In other words, we have shown that the power series P(x) and Q(x) satisfy \frac{P(x)}{Q(x)} = \tan x = \frac{\sin x}{\cos x} and therefore P(x) = k \sin x and Q(x) = k \cos x for some k. The observation that Q(0) = 1 = \cos 0 (or our earlier observation that P(x)^2 + Q(x)^2 = 1 for all x) gives k = 1, thereby showing that P(x) = \sin x and Q(x) = \cos x.

So much for \sin x and \cos x. Just as an aside, observe that if we take i to be a symbol satisfying i^2 = -1, then
\displaystyle \begin{aligned}  \cos x + i\sin x   &= Q(x) + iP(x) \\  &= \left(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots\right) + i\left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots \right) \\  &= 1 + ix + \frac{-x^2}{2!} + \frac{-ix^3}{3!} + \frac{x^4}{4!} + \frac{ix^5}{5!} + \frac{-x^6}{6!} + \frac{-ix^7}{7!} + \dots \\  &= 1 + ix + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} + \frac{(ix)^5}{5!} + \frac{(ix)^6}{6!} + \frac{(ix)^7}{7!} + \dots  \end{aligned}
the right hand side of which looks very much like the result of “substituting” y = ix in the known (real) power series
\displaystyle e^y = 1 + y + \frac{y^2}{2!} + \frac{y^3}{3!} + \dots
(which itself can be proved using the term-wise differentiation above and the defining property \frac{d}{dx} e^x = e^x, say).

So this is one heuristic justification for us to define e^{ix} = \cos x + i\sin x.
Or, if we define e^{ix} as the result of substituting ix in the real power series for e^y, this proves that e^{ix} = \cos x + i\sin x.

Written by S

Fri, 2013-03-08 at 00:33:12

Posted in mathematics

A better keyboard layout for typing IAST on Mac OS X (based on EasyUnicode)

with 13 comments

To type IAST (English letters with diacritics, for Sanskrit transliteration) on Mac OS X, perhaps the easiest way, rather than to use transliteration tools, is to get a keyboard layout that does it. Just to be clear, this is the alphabet we want:

a ā i ī u ū ṛ ṝ ḷ ḹ e ai o au ṃ ḥ
k kh g gh ṅ
c ch j j ñ
ṭ ṭh ḍ ḍh ṇ
t t d dh n
p ph b bh m
y r l v ś ṣ s h

In other words, the special characters needed are:

  • Letters with macron above: ā ī ū ṝ ḹ plus it may be occasionally useful to have ē and ō as well
  • Letters with dot below: ṭ ḍ ṇ ṣ (the retroflex consonants), also the vowels ṛ ṝ ḷ ḹ, plus ṃ and ḥ (anusvāra)
  • Letters with other marks above: ṅ ñ ś

There is a keyboard layout that does this: It’s called “EasyUnicode”, created by Toshiya Unebe (Nagoya University), and is documented at (“EasyUnicode version 5” it says) (PDF version), and you can download it from (EBMP) (=Early Buddhist Manuscripts Project, University of Washington) or also “Pali Fonts for PC and Unicode”. (Page in Japanese.)

This keyboard layout is just like the usual (US English) layout ordinarily, but when you hold down the Alt (Option) key and press a, you get ā, similarly Option+s gives ś, Option+n gives ñ and Option+g gives ṅ, etc. The full mapping is available along with other documentation in the download above.

This is very convenient. One issue with the layout is that also overrides a lot of keys for no apparent reason (Ctrl-A / Ctrl-E etc. stopped working for me), so I got Ukelele from SIL, and wrote my own keyboard layout. I’ve called it EasyIAST, and it is available here for now (NOTE: do not simply click on the link or your browser will try to render as xml and probably display an error; you have to download instead). I plan to add a README etc. and distribute it in some proper way later; for now you can use the instructions from EasyUnicode above. If you find it useful and/or make any improvements, please let me know as well.

If some time is available, it would be good to make a Devanagari keyboard layout along the same lines.

Written by S

Tue, 2013-01-22 at 11:28:55

Posted in sanskrit

Tagged with , ,

The potions puzzle by Snape in Harry Potter and the Philosopher’s Stone

with 3 comments

Sometime this week, I reread the first Harry Potter book (after at least 10 years… wow, has it been that long?), just for contrast after reading Rowling’s adult novel The Casual Vacancy (on which more later). Anyway, in the penultimate chapter there is a puzzle:

He pulled open the next door, both of them hardly daring to look at what came next — but there was nothing very frightening in here, just a table with seven differently shaped bottles standing on it in a line.
“Snape’s,” said Harry. “What do we have to do?”
They stepped over the threshold, and immediately a fire sprang up behind them in the doorway. It wasn’t ordinary fire either; it was purple. At the same instant, black flames shot up in the doorway leading onward. They were trapped.
“Look!” Hermione seized a roll of paper lying next to the bottles. Harry looked over her shoulder to read it:

Danger lies before you, while safety lies behind,
Two of us will help you, whichever you would find,
One among us seven will let you move ahead,
Another will transport the drinker back instead,
Two among our number hold only nettle wine,
Three of us are killers, waiting hidden in line.
Choose, unless you wish to stay here forevermore,
To help you in your choice, we give you these clues four:
First, however slyly the poison tries to hide
You will always find some on nettle wine’s left side;
Second, different are those who stand at either end,
But if you would move onward, neither is your friend;
Third, as you see clearly, all are different size,
Neither dwarf nor giant holds death in their insides;
Fourth, the second left and the second on the right
Are twins once you taste them, though different at first sight.

I became curious about whether this is just a ditty Rowling made up, or the puzzle actually makes sense and is consistent. It turns out she has constructed it well. Let’s take a look. This investigation can be carried out by hand, but we’ll be lazy and use a computer, specifically Python. The code examples below are all to be typed in an interactive Python shell, the one that you get by typing “python” in a terminal.

So what we have are seven bottles, of which one will take you forward (F), one will let you go back (B), two are just nettle wine (N), and three are poison (P).

>>> bottles = ['F', 'B', 'N', 'N', 'P', 'P', 'P']

The actual ordering of these 7 bottles is some ordering (permutation) of them. As 7 is a very small number, we can afford to be inefficient and resort to brute-force enumeration.

>>> import itertools
>>> perms = [''.join(s) for s in set(itertools.permutations(bottles))]
>>> len(perms)

The set is needed to remove duplicates, because otherwise itertools.permutations will print 7! “permutations”. So already the number of all possible orderings is rather small (it is \frac{7!}{2!3!} = 420). We can look at a sample to check whether things look fine.

>>> perms[:10]

Now let us try to solve the puzzle. We can start with the first clue, which says that wherever a nettle-wine bottle occurs, on its left is always a poison bottle (and in particular therefore, a nettle-wine bottle cannot be in the leftmost position). So we must restrict the orderings to just those that satisfy this condition.

>>> def clue1(s): return all(i > 0 and s[i-1] == 'P' for i in range(len(s)) if s[i]=='N')
>>> len([s for s in perms if clue1(s)])

(In the code, the 7 positions are 0 to 6, as array indices in code generally start at 0.)
Then the second clue says that the bottles at the end are different, and neither contains the potion that lets you go forward.

>>> def clue2(s): return s[0] != s[6] and s[0] != 'F' and s[6] != 'F'
>>> len([s for s in perms if clue1(s) and clue2(s)])

The third clue says that the smallest and largest bottles don’t contain poison, and this would be of help to Harry and Hermione who can see the sizes of the bottles. But as we readers are not told the sizes of the bottles, this doesn’t seem of any help to us; let us return to this later.

The fourth clue says that the second-left and second-right bottles have the same contents.

>>> def clue4(s): return s[1] == s[5]
>>> len([s for s in perms if clue1(s) and clue2(s) and clue4(s)])

There are now just 8 possibilities, finally small enough to print them all.

>>> [s for s in perms if clue1(s) and clue2(s) and clue4(s)]

Alas, without knowing which the “dwarf” and “giant” bottles are, we cannot use the third clue, and this seems as far as we can go. We seem to have exhausted all the information available…

Almost. It is reasonable to assume that the puzzle is meant to have a solution. So even without knowing where exactly the “dwarf” and “giant” bottles are, we can say that they are in some pair of locations that ensure a unique solution.

>>> def clue3(d, g, s): return s[d]!='P' and s[g]!='P'
>>> for d in range(7):
...   for g in range(7):
...     if d == g: continue
...     poss = [s for s in perms if clue1(s) and clue2(s) and clue4(s) and clue3(d,g,s)]
...     if len(poss) == 1:
...       print d, g, poss[0]

Aha! If you look at the possible orderings closely, you will see that we are down to just two possibilities for the ordering of the bottles.
Actually there is some scope for quibbling in what we did above: perhaps we cannot say that there is a unique solution determining the entire configuration; perhaps all we can say is that the puzzle should let us uniquely determine the positions of just the two useful bottles. Fortunately, that gives exactly the same set of possibilities, so this distinction happens to be inconsequential.

>>> for d in range(7):
...   for g in range(7):
...     if d == g: continue
...     poss = [(s.index('F'),s.index('B')) for s in perms if clue1(s) and clue2(s) and clue4(s) and clue3(d,g,s)]
...     if len(set(poss)) == 1:
...       print d, g, [s for s in perms if clue1(s) and clue2(s) and clue4(s) and clue3(d,g,s)][0]

Good. Note that there are only two configurations above. So with only the clues in the poem, and the assumption that the puzzle can be solved, we can narrow down the possibilities to two configurations, and be sure of the contents of all the bottles except the third and fourth. We know that the potion that lets us go forward is in either the third or the fourth bottle.

In particular we see that the last bottle lets us go back, and indeed this is confirmed by the story later:

“Which one will get you back through the purple flames?”
Hermione pointed at a rounded bottle at the right end of the line.
She took a long drink from the round bottle at the end, and shuddered.

But we don’t know which of the two it is, as we can’t reconstruct all the relevant details of the configuration. Perhaps we can reconstruct something with the remaining piece of information from the story?

“Got it,” she said. “The smallest bottle will get us through the black fire — toward the Stone.”
Harry looked at the tiny bottle.
Harry took a deep breath and picked up the smallest bottle.

So we know that the bottle that lets one move forward is in fact in the smallest one, the “dwarf”.

>>> for d in range(7):
...   for g in range(7):
...     poss = [s for s in perms if clue1(s) and clue2(s) and clue4(s) and clue3(d,g,s)]
...     if len(poss) == 1 and poss[0][d] == 'F':
...       print d, g, poss[0]

This narrows the possible positions of the smallest and largest bottles (note that it says the largest bottle is one that contains nettle wine), but still leaves the same two possibilities for the complete configuration. So we can stop here.

What we can conclude is the following: apart from the clues mentioned in the poem, the “dwarf” (the smallest bottle) was in either position 2 (third from the left) or 3 (fourth from the left). The biggest bottle was in either position 1 (second from the left) or 5 (sixth from the left). With this information about the location of the smallest bottle (and without necessarily assuming the puzzle has a unique solution!), Hermione could determine the contents of all the bottles. In particular she could determine the location of the two useful bottles: namely that the bottle that lets you go back was the last one, and that the one that lets you go forward was the smallest bottle.

>>> for (d,g) in [(2,1), (2,5), (3,1), (3,5)]:
...   poss = [s for s in perms if clue1(s) and clue2(s) and clue4(s) and clue3(d, g, s)]
...   assert len(poss) == 1
...   s = poss[0]
...   assert s.index('B') == 6
...   assert s.index('F') == d
...   print (d,g), s
(2, 1) PNFPPNB
(2, 5) PNFPPNB
(3, 1) PNPFPNB
(3, 5) PNPFPNB

It is not clear why she went to the effort to create a meaningful puzzle, then withheld details that would let the reader solve it fully. Perhaps some details were removed during editing. As far as making up stuff for the sake of a story goes, though, this is nothing; consider for instance the language created for Avatar which viewers have learned.

See also which does it by hand, and has a perfectly clear exposition (it doesn’t try the trick of guessing that solution is unique before reaching for the additional information from the story).

Written by S

Mon, 2012-10-22 at 02:00:12

Translating metaphor into English: Time and Motion?

with 5 comments

In a book called A History of Kanarese Literature, by Edward Rice (1921), he makes the following comment (p. 106):

The other is that a Kanarese poem defies anything like literal translation into another language. To give any idea of the spirit of the original it would be necessary to paraphrase freely, to expand the terse and frequent metaphors into similes, and to give a double rendering of many stanzas. An example will make this clear. The opening stanza of the Jaimini Bharata is given in Sanderson’s translation as follows:

May the moon-face of Vishnu, of Devapura, always suffused with moonlight smile, full of delightful favour-ambrosial rays—at which the chakora-eye of Lakshmi is enraptured, the lotus-bud heart of the devout expands, and the sea of the world’s pure happiness rises and overflows its bounds—give us joy.

The following is an attempt, by means of a freer rendering, to retain something of the spirit of the original:

When the full moon through heaven rides,
Broad Ocean swells with all its tides ;
The lotus blossom on the stream
Opens to drink the silv’ry beam ;
And far aloft with tranced gaze
The chakor bird feeds on the rays.

So, when great Vishnu’s face is seen,—
Whom men adore at Devapore—
Like to the sea, the devotee
Thrills with a tide of joy ;
Like to the flower, that blissful hour
The heart of the devout expands ;
And Lakshmi Queen, with rapture keen,
Watches with ever-radiant face
For her great Consort’s heavenly grace.
O may that grace be ours !

I’m wondering about this change. Apart from the versification—you know, being an actual poem instead of stilted prose—when it comes to just the idea, is it better? Why? How? Is it more readable? More understandable? Most importantly, does this change better “retain the spirit of the original”?

[Aside: just to be mischievous, we can with the wonders of technology do the following:

moon moonlight rays chakora bird lotus sea
Vishnu’s face smile his grace Lakshmi’s eye heart of the devout world’s happiness

to ruin the poem.]

For one thing, he has changed the metaphor (rūpaka) of the original into simile (upamā).
Probably the reason is that the compressed quality of the original, a prominent characteristic of Sanskrit and other classical Indian literature, is unsuitable for English, whose readers are typically unprepared for it. Is there more to it? Is this a general difference between the two literary cultures?

I’m wondering all this because Daniel Ingalls says something along similar lines in his honestly-written general introduction “Sanskrit poetry and Sanskrit Poetics” (from his translation of the Subhāṣita-ratna-kośa anthology):

As a result, Sanskrit is lacking in what is perhaps the chief force of English poetry: its kinesthetic effect. What I mean can be shown by an old ballad:

Martinmas wind, when wilt thou blow
and shake the green leaves off the tree…

One can feel the leaves shaking, and one shivers in the next line to the “Frost that freezes fell / and blowing snow’s inclemency.” One can find verses that produce this muscular effect in Bengali, and although I cannot speak at first hand of other modern Indian literatures, I imagine that one can find the effect in them as well. But it is only rarely that one finds it in Sanskrit. The powers of Sanskrit are of a different order.
[The following verse] is by Yogeśvara, an excellent poet who is capable of better things. In it he uses a strikingly elaborate metaphor:

Now the great cloud-cat,
darting out his lightning tongue,
licks the creamy moon
from the saucepan of the sky.

The effect here is gained by intellectual, entirely rational means. The metaphor is complete in every detail: cat, tongue, cream, and saucepan—cloud, moon, lightning, and sky. It is almost like an exercise from a manual of logic under the chapter “Analogy.” Compare the verse with a well-known passage of T. S. Eliot which uses several similar ideas, but uses them very differently:

The yellow fog that rubs its back upon the window-panes
The yellow smoke that rubs its muzzle on the window-panes,
Licked its tongue into the corners of the evening,
Lingered upon the pools that stand in drains, …

This from one who is often called an intellectual poet. And yet Eliot gets his effect in every line from the irrational, the strong but imprecise memory we have of fog and cats, the childhood associations of certain words and idioms. Consider the line: “Licked its tongue into the corners of the evening.” It brings to sudden flower certain homely and completely natural phrases: “licks his tongue around the bowl,” or “licks his tongue into the corner of the dish.” The idiom is suddenly transfigured by bringing it into juxtaposition with the last three words, “of the evening.” This transfiguration of language becomes impossible without a natural-language basis.

Is there a general point here that English poetry uses vague, fuzzy, but “kinesthetic” effects where Sanskrit (or classical Indian) poetry uses compressed metaphors that paint a precise and detailed picture? I think there is some merit to the idea that, by and large, Sanskrit poetry is “static”, not “dynamic”. It is not a stream in motion; it hasn’t any “flow”. It is more a pearl in itself, that dazzles as you read. If poetry is imagination and the evocation of something other-worldly, it seems to me that Sanskrit poetry in general / at its best, conjures a world that one can calmly dwell in for a while, not an evocative fleeting idea that escapes as you try to grasp it, one which has appeal more in the chasing. Consider the importance accorded ultimately to stability / sthāyī-bhāva in all Indian arts, from poetry to theatre to dance.

This requires more thought and elaboration, but one may as well quote the final lines of Ingalls’s introduction (emphasis mine):

One may argue today, as the Sanskrit critics argued in the past, the relative importance of the various factors of Sanskrit verse which I have discussed. Vocabulary, grammar, meter: these are all necessary. Figures of speech, both verbal and intellectual, furnish delight. Mood is what is sought, though the grand successes of Sanskrit I would say go beyond mood to a sort of universal revelation, to what James Joyce, drawing on the vocabulary of religion, called an epiphany. To achieve this success impersonality is a prerequisite and suggestion is the chief instrument. If I were to single out for admiration one factor above the others in this complex it would be suggestion, not because it is unknown in other languages but because the Sanskrit poets use it with such brilliance and because it seems to me the most intimately connected of all the factors with the excitement, the sudden rushing of the mind into a delightful, calm expansion, that one occasionally derives from Sanskrit poetry and that brings one who has once known it constantly back for further draughts.

Written by S

Sat, 2012-09-29 at 13:09:01

Bill Thurston

with 2 comments

Somehow I am saddened to hear of the death of William Thurston, even though I never knew him nor can I even understand his work.

For years I have been pointing people at this article of his, which I find illuminating and inspiring, even in ways that have nothing to do with mathematics:
5-paragraph version:
Riffs on his “derivative” exercise:

User profile (
“Mathematics is a process of staring hard enough with enough perseverance at at the fog of muddle and confusion to eventually break through to improved clarity. I’m happy when I can admit, at least to myself, that my thinking is muddled, and I try to overcome the embarrassment that I might reveal ignorance or confusion. Over the years, this has helped me develop clarity in some things, but I remain muddled in many others. I enjoy questions that seem honest, even when they admit or reveal confusion, in preference to questions that appear designed to project sophistication.” :
“At that time, I prided myself in reading quickly. I was really amazed by my first encounters with serious mathematics textbooks. I was very interested and impressed by the quality of the reasoning, but it was quite hard to stay alert and focused. After a few experiences of reading a few pages only to discover that I really had no idea what I’d just read, I learned to drink lots of coffee, slow way down, and accept that I needed to read these books at 1/10th or 1/50th standard reading speed, pay attention to every single word and backtrack to look up all the obscure numbers of equations and theorems in order to follow the arguments.” :
“When listening to a lecture, I can’t possibly attend to every word: so many words blank out my thoughts. My attention repeatedly dives inward to my own thoughts and my own mental models, asking ‘what are they really saying?’ or ‘where is this going?’. I try to shortcut through my own understanding, then emerge to see if I’m still with the lecture. It’s the only way for me, and it often works.”
“My first tenure-track job interview was at Cornell. During and after my job talk, most people were pretty quiet, but there was one guy who kept asking very penetrating and insightful questions. And it was very confusing, because I knew all the number theorists at Cornell, and I had no idea who this guy was, or how it was that he obviously understood my talk better than anybody else in the room, possibly including me.”
“When I grew up I was a voracious reader. But when I started studying serious mathematical textbooks I was surprised how slowly I had to read, at a much lower speed than reading non-mathematical books. I couldn’t just gloss over the text. The phrasing and the symbols had been carefully chosen and every sign was important. Now I read differently. I rarely spend the time and effort to follow carefully every word and every equation in a mathematics article. I have come to appreciate the importance of the explanation that lies beneath the surface. […] I have little expectation for the words to be a good representation of the real ideas. I try to tunnel beneath the surface and to find shortcuts, checking in often enough to have a reasonable hope not to miss a major point. I have decided that daydreaming is not a bug but a feature. If I can drift away far enough to gain the perspective that allows me to see the big picture, noticing the details becomes both easier and less important.
I wish I had developed the skill of reading beneath the surface much earlier. As I read, I stop and ask, What’s the author trying to say? What is the author really thinking (if I suppose it is different from what he put in the mathematical text)? What do I think of this? I talk to myself back and forth while reading somebody else’s writing. But the main thing is to give myself time, to close my eyes, to give myself space, to reflect and allow my thoughts to form on their own in order to shape my ideas.” :
“A prominent mathematician once remarked to me that Thurston was the most underappreciated mathematician alive today. When I pointed out that Thurston had a Fields medal and innumerable other accolades, he replied that this was not incompatible with his thesis.”

He believed that this human understanding was what gave mathematics not only its utility but its beauty, and that mathematicians needed to improve their ability to communicate mathematical ideas rather than just the details of formal proofs.

Benson Farb, a mathematician at the University of Chicago and a student of Thurston, said in an email, “in my opinion Thurston is underrated: his influence goes far beyond the (enormous) content of his mathematics. He changed the way geometers/topologists think about mathematics. He changed our idea of what it means to ‘encounter’ and ‘interact with’ a geometric object. The geometry that came before almost looks like pure symbol pushing in comparison.”

Reactions like this are hard to explain: About the Proof&Progress article, says:

“Reading Thurston’s response was one of the most stirring intellectual experiences of my life. It truly struck a cord with my own conception of mathematics. To me, it has the status that the Declaration of Independence has to many Americans, or the U.N. charter has to other global citizens.”

See these for a better summary:

Edit [2012-09-28]: He has also done work in computer science! A fundamental result in data structures (on the number of rotations needed to transform one binary tree into another) was proved in a paper by Sleator–Tarjan–Thurston (with 278 citations). According to DBLP, he has three STOC papers.

See also: (“Mathematical Education”)

Edit [2016-02-20]: A collection of reminiscences about Thurston, in the Notices of the AMS, issues December 2015 and January 2016.

Written by S

Fri, 2012-08-24 at 15:28:19

Posted in mathematics