# The Lumber Room

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

## The idea of logarithms, and the first appearance of e

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 +05:30

Posted in mathematics

## Wodehouse on Conan Doyle

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.

and

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.

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.

Sources:

http://bullyscomics.blogspot.in/2008/01/wodehouse-week-wodehouses-introduction.html

http://bakerstreetbeat.blogspot.in/2012/01/pg-wodehouse-fan-for-life.html

http://bakerstreetbeat.blogspot.in/2013/03/a-master-humorist-takes-on-sherlock.html

http://plausive.dreamwidth.org/519931.html

http://fc09.deviantart.net/fs70/f/2012/287/1/2/reprint__introduction_by_p_g__wodehouse_by_chaosfive55-d5hsw78.html

https://secure.flickr.com/photos/littlestuffedbull/2173532528/

https://www.lib.umn.edu/pdf/holmes/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” (http://thenostalgialeague.com/olmag/dudley-jones.html), in Punch on April 29, 1903 (http://madameulalie.org/punch/Dudley_Jones_1.html) and May 6, 1903 (http://madameulalie.org/punch/Dudley_Jones_2.html)

* Wodehouse wrote (unsigned) a poem called “Back to his Native Strand” for Punch on May 27, 1903: http://madameulalie.org/punch/Back_to_his_native_Strand.html

* Wodehouse did an “interview” of ACD in “VC” magazine, July 2, 1903: http://madameulalie.org/vc/Grit.html

* Wodehouse wrote (unsigned) “The Prodigal” for Punch on September 23, 1903: http://madameulalie.org/punch/The_Prodigal.html

Written by S

Fri, 2013-10-11 at 08:37:03 +05:30

Posted in literature

## Cricket poems

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,
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,

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.

Missed

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;
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.

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

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

Written by S

Sun, 2013-09-01 at 14:35:50 +05:30

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

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.

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 +05:30

Posted in language, literature, quotes

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

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.)

Written by S

Mon, 2013-04-08 at 11:24:08 +05:30

Posted in mathematics

## Trajectory of a point moving with acceleration perpendicular to velocity

(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 +05:30

Posted in mathematics

## Typing Kannada on Mac OS X

with one comment

(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, and turn on MacUIM. Tick “Show Input menu in menu bar” too.
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.
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 +05:30

Posted in compknow