June 21, 2006

Shoulders of giants

On an otherwise nondescript day in the year 212 BC, in a city called Syracuse, an invading Roman soldier came upon a 75 year old man. "Follow me to my commander, Marcellus!" the soldier ordered. The old man was closely examining a diagram dotted with circles, oblivious to his surroundings. He refused to go till he had finished solving his problem.

"Do not touch my circles!" he scolded the soldier. That enraged the soldier so much that he drew his sword and killed the old man.

Fortunately, history has forgotten the soldier's name. The man he killed that day was Archimedes.

As far as Rome was concerned, killing Archimedes was a glorious triumph. For the previous three years, Syracuse, then part of the Greek empire, had held off the Roman might with something never seen before: scientific engineering. The Syracusans knew a lot about levers and pulleys and catapults, and they put these instruments to clever use. Grapnels lowered from huge cranes caught the bows of the Roman ships and hoisted them up using pulleys, spilling Rome's proud sailors into the sea. On land, catapulted balls and darts and bolts showered down on Marcellus's men, sending them fleeing in panic.

With techniques like that, Syracuse was showing Rome that brains could beat brawn.

And it was Archimedes who discovered the secrets that enabled Syracuse to build its ingenious machines. It took a three-year siege for Marcellus to finally overcome Syracuse, and for one of his soldiers to kill Archimedes.

One small triumph for Rome, sure. But one giant tragedy for mankind. For Archimedes of Syracuse was a truly brilliant mathematician, physicist and engineer.

He was the first thinker to apply mathematics consistently to physics, so becoming the pioneer of physics as a science. And in searching for general principles to apply to specific engineering problems, he was the first scientific engineer.

And perhaps above all, Archimedes was a champion of method. His results were marvellous in themselves, but he was even more concerned with how he got those results. His method had to be unshakeable. Starting with simple postulates, he used unimpeachable logic to derive his propositions.

The best known of those is Archimedes' Law: an object placed in water displaces an amount of water whose volume equals that of the object. (Well, there's a bit more to it, but that's the gist of it). The king of Syracuse once asked Archimedes to find out whether, as he suspected, his crown was not pure gold. Legend tells us that Archimedes was thinking about the problem while taking a bath one day. He noticed that the water level rose as he sank into the bath. Suddenly realizing the answer, he ran naked through the streets to tell the king of his discovery, shouting "Eureka!" ("I have found it!").

Which may just be the most famous public display of nudity for scientific purposes in history.

Today it is amusing to read of the fiendish machines Archimedes devised to defend Syracuse. They couldn't have been very amusing to the Romans, though. Fed on the virtues of simply overwhelming the enemy with brute force, they were baffled by what happened at Syracuse. Which previous enemy had been as contemptuous of Roman military strength as to lift her ships up and toss her sailors into the sea?

To Archimedes, though, the machines were mere applications of his principles: the lever, the pulley and others. And just as significant to him as the victories of his machines were his other applications of those principles.

For example, he used the principle of the lever to derive the volume of a sphere. His elegant method considered how differing circles balanced each other on an imaginary lever, so deducing their weights and volumes. He went on to use the same method to find the volume of other objects of mathematical interest, like the ellipsoid. In fact, Archimedes was so fond of this method that, at his request, his tombstone was engraved with a sphere in a cylinder, illustrating his method.

Like other thinkers of his time, Archimedes also gave a lot of thought to calculating pi, the ratio of the circumference of a circle to its diameter. Now this was an interesting pursuit because, of course, pi cannot be expressed as a fraction, a ratio between two numbers -- so it is impossible to state an exact value for it. It startled the Greeks that as simple an object as a circle contained within it the diabolical irrationality of pi.

Archimedes, characteristically less concerned with the value than with how to find it, was the first to find a method to calculate pi to any desired degree of accuracy. Not just that, it was 19 centuries before his method was bettered. In this, as in his other pursuits, Archimedes stood head and shoulders above his contemporaries. And above most who came afterward as well.

So if an ignorant Roman killed the man in 212 BC, his ideas and his influence on mathematics live on. Not even the might of a great Empire could kill those.

For you might say, it was Archimedes who first showed the world that the thinkers could defeat the thugs.

29 comments:

Anonymous said...

From

http://www.math.nyu.edu/~crorres/
Archimedes/Death/Histories.html

where you can find a lot of other good stuff.

Alfred North Whitehead (1861-1947), An Introduction to Mathematics, Williams & Norgate, London, 1911, (Oxford University Press, 1958, pages 25-26)

The death of Archimedes by the hands of a Roman soldier is symbolical of a world-change of the first magnitude: the Greeks, with their love of abstract science, were superseded in the leadership of the European world by the practical Romans. Lord Beaconsfield, in one of his novels, has defined a practical man as a man who practises the errors of his forefathers. The Romans were a great race, but they were cursed with the sterility which waits upon practicality. They did not improve upon the knowledge of their forefathers, and all their advances were confined to the minor technical details of engineering. They were not dreamers enough to arrive at new points of view, which could give a more fundamental control over the forces of nature. No Roman lost his life because he was absorbed in the contemplation of a mathematical diagram.

Ravi

Vulturo said...

Beautiful post, Dilip

In The Shadows said...
This comment has been removed by a blog administrator.
In The Shadows said...

http://retributions.wordpress.com/2006/06/16/why-we-must-sing-vande-mataram/#comment-1819

Dilip, we are missing ya :)

In The Shadows said...

Dilip, btw, you said -

an object placed in water displaces an amount of water equal in weight to the object itself.

Which is totally wrong.

Basic logic . Say you submerge an inflated balloon in water. Will it displace water equal to the weight of the balloon ???

Help yourself!

Anonymous said...

intheshadows, I understand your insatiable desire to bait Dilipp. but can you please at least stick to the subject?!

Indian said...

Sadly it is the thugs that call the shots.
At present there are many Archimedes in our midst. There is a thug Arjun Singh who is hell bent on destroying all learning in the country.
In fact his actions are worse - almost equivalent to destroying the Nalanda Universities.

I pray for our civilisation to survive the likes of Arjun Singh and all of you who invite others to invade the country and have become his shameless cheerleaders -Uma, Albert and the like

Boskoe said...

Just when I think that there is no way I will read one more post of yours (because I dont agree with your opinions on most of them), you go ahead and write a beautiful piece like this one.

Good one! Keep it up! ...and in spite of what i said earlier, I do intend to keep reading your posts -call it masochism! :-)

Dilip D'Souza said...

Damn, I can't believe I made that simple mistake. In the Shadows is right, of course. An object placed in water displaces an amount of water equal in volume to the object itself.

Thanks much for the correction, In the Shadows. I'm making it in the text as soon as I put this comment up.

Ravi, thanks for that pointer and quote. that's an interesting view of the transition from the Greeks to the Romans. Oddly enough, it reminds me of the fascinating explanation for the Abel/Cain story in "Ishmael".

Boskoe, thank you. Please stop by any time. Vulturo, thanks pal. You're kind.

km said...

Dear Sir,

Your post, like they say in these parts, rocks. Like an Archimedes' Screw.

Anonymous said...

>> I pray for our civilisation to survive the likes of Arjun Singh and all of you who invite others to invade the country and have become his shameless cheerleaders -Uma, Albert and the like

You, yes you the Indian, you certainly have a gall to invoke the names of Nalanda - the premier hindu school of learning!!

Please pay close attention buddy, our reverence to distant Roman or Greek scholars comes first and foremost ;-)

anon June 21, 2006 8:01 PM
but can you please at least stick to the subject?!
'Sticking it' to the subject is Dilip's speciality and doubt shadows (or anyone for that matter) can match that.

Anindo said...

Hmm!

Is not supporting men like Archimedes "elitism"? What is the need for these ideas when people are dying on the streets?

Did Archimedes make the life of the poor in Syracuse more comfortable by his new weapons? Would it not have been better if money spent in funding Archimedes' grand ideas could have been spent to eradicate poverty in Syracuse?

Did he protest against the war running naked in the streets? No, he celebrated his ideas by running naked on the streets. What a jerk!

Did Archimedes give credit for his ideas to his fellow inventors because of social justice? Imagine the state of mind of the lesser mortals who knew that they could not come up with these ideas and demand that Archimedes share his fame with them. Everybody deserves equal treatment irrespective of their merit in the society. Am I wrong here?

Would Archimedes agree to sacrifice his more deserving place as an inventor, a Physicist, and a Mathematician in the society for lesser mortals who faced persecution in the society earlier in the history?

If yes, would we have celebrated Archimedes' ideas today? If no, was not Archimedes arrogant and a man devoid of morality - a person who looks after his own interest?

I wonder!

Regards,

c2c said...

Vey interesting post. Its always amazing to read about those who were centuries ahead of their fellow men, like Leonardo Da Vinci or the ancient Hindu vedics.

srinivas said...

Sheesh!

Can't people read an interesting post about a most fascinating person without discussing Dilip's political belief's? Get a life.

Apprently, Archimedes is also supposed to have designed and used giant concave mirrors that focused the sun's rays on invading Roman ships causing them to go up in flames.

confused said...

@Dilip,

Rocking post. Much knowledge was gained.

Compliments.

@anonymous at 10.34 pm.

Why in the name of IPR should the name of Nalanda University not be invoked? Because it was a Hindu school? Ah! So let us demolish Taj Mahal because it is a Muslim place and stop taking the name of Ashoka because he was a Buddhist....

I could go on all day..but I have work to do.

best

Anonymous said...

Karthik:

>> Its always amazing to read about those who were centuries ahead of their fellow men, like Leonardo Da Vinci or the ancient Hindu vedics.

Ah, ah..ahem... 'ancient Hindu vedics' date before Archimedes 212 BC - just an FYI. Even Kautialya was writing Arthashastra by then ;)

Confused: Don't be so quick to pull the trigger, try to read what's being said. Dilip get's it, why don't you?



Srinivas:
Can't people read an interesting post about a most fascinating person without discussing Dilip's political belief's? Get a life.
I think you miss the whole point here. No one, not one person here has put himself in the shoes of that soldier who had to commit such a vile act.

This elite Archimedes could frolick around naked and do his math sums on street, come on, if he's not an elitist what his he? A typist? Heard anyone else during that time frame chiesling poems on busy streets?

If soldier hadn't done what he did he too could have been killed by Marcellus himself? What would have happened his wife and kids? Any ideas as to how many eggs they'd have to survive on? Read Battle of Thermopyle (spelling) to get an idea of some ancient soldiers rations. Their kids would have been on street since they didn't have reservations for the soldier class in Rome then. They didn't have travel subsidy to visit their pagan gods either - poor souls. Add to this that these soliders had to scream at the top of their lungs 'hail ceaser' every day to prove their loyalty without any fatwa coverage..... boy those soldiers had a though time then. Don't trivilize it please.

barbarindian said...

If I am not mistaken this stuff is taught in history classes in high school. Perhaps I am missing the point of this post but it appears to be clearly political. Just like the Yimou Zhang movie Hero which of course reinforces the point that an individual must subordinate his self-interests to that of the state.

It may be prudent to remind Dilip that there are other important lessons one can learn from history for he is clearly ignoring some of them, for instance:

http://www.state.gov/r/pa/ho/time/dr/17672.htm

I think Dilip is comparing himself to Archimedes and the anti-liberals as thugs. Who knows, but we certainly hope he takes regular baths in a bathtub or otherwise and more importantly does not run around naked in a moment of inspiration - that would be scary.

Happy-Go-Lucky said...

Nice write up Dilip.
I like reading/watching documentaries on the great lives from history.

I also read somewhere that Archimedes once used convex mirror to reflect sunlight onto the sails of approaching enemy ships and burnt them down. Great idea, but dont know how true it is.

Anonymous said...

Offered without comment:

Archimedes' practical inventions did make the life of the aam aadmi easier. In particular he devised a "screw" for lifting water from streams. Presumably, the ability to measure the purity of gold and other substances was of use to society at large, even if, in the first instance, it was used to levy the death penalty on the cheating goldsmith (this is a dispute between the bourgeois and ruling classes and, as such, should not concern the Marxists amongst us, except that such internal strife weakens the feudal-capitalist classes).

Archimedes was, by some accounts, related to the Syracuse ruling family and definitely part of the Syracusean elite.

Archimedes himself scorned his practical inventions (although, that didn't stop him from inventing on a scale since unmatched), valuing his theoretical researches more highly. Incidentally he discovered both the integral and differential calculus for himself - he could compute the volumes of rotated conics and draw tangents to pretty much any curve you could describe sensibly without the aid of analytic geometry. The Fundamental Theorem of Calculus, however, seems to have eluded him.

Archimedes' Law: The loss of weight in water is equal to the weight of the water displaced.

Archimedes was almost certainly the greatest thinker of all time. If Europe had followed his intellectual lead (as it eventually and belatedly did) rather than that of Aristotle and Plato, physics and mathematics would have advanced by 1800 years.

Ravi

Vulturo said...

Barbarindian,

I'm afraid you are reading too much into this or something.

To me, the piece comes across as a wonderful post. A refreshing breather, in fact, from all the bickering which usually happens in the blogosphere (and which we all participate in).

There's no sense in arguing or acting confrontationist, just for the heck of it. It makes you look stupid.

Dilip D'Souza said...

Ravi, physics and mathematics would have advanced by 1800 years.

A most intriguing claim! If I'm calculating this right, you're saying that Archimedes' physics and mathematics were essentially modern-day in their character? Ahead of Gauss and Galileo and Euler and Fermat and so many others? Please do elaborate.

Srinivas and H-Go-L, The story with the mirrors is another famous Archimedes one. But I've always been a little sceptical of it. Why for example, would those ships have simply stood still and waited to be burned to a crisp?

karthik, thanks.

km, confused: when you say "rock", I'm tempted to reply, "-n-roll, Signora Castoroili!" But I won't, promise.

Irrelevant aside: as kids, we had a pet tortoise called Archimedes.

Anonymous said...

>> There's no sense in arguing or acting confrontationist, just for the heck of it. It makes you look stupid.

Well by that logic, Dilip should the king of stupidity?

In The Shadows said...

Dilip, thats what is merit.

An engineer manning a power plant should not forget the laws of thermodynamics. And neither should a marine architect or a engineer working on hydraulics and pneumatics forget the Archimedes principle.

Of course, nobody is perfect. Mistakes do happen. The point here is that such basic things should not be forgotten. But in critical situations, one cannot afford to forget the basic laws.

This is what is merit Dilip, which you fail to understand. A doctor cannot forget the locations of the major veins and arteries connected to the heart, can he ? LIke you forgot something as simple as Archimedes principle.

I wonder what will happen if someone forgot the bernoulli principle while building an aircraft?? Or what if an engineer in a nuclear power plant maintaining pipings for steam forgets the venturi principle.

Dilip, I hope you now realize what is merit. Its not about caste, its about selection. And regardless of caste.

Anonymous said...

Dilip, your calculation is slightly off. I chose "1800" with some care. Eighteen hundred years after the death of Archimedes brings us to 1588 - just before the grand age ushered in by Descartes, Fermat and Pascal- the year Sir Frances Drake defeated the Spanish Armada, incidentally (this last has absolutely no significance whatsoever, but I couldn't help showing off!).

Archimedes was definitely "modern" in thought. Indeed, it has been said (by someone of consequence whose name I forget) that Archimedes would probably have been the best person to understand Einstein's theory of relativity (mark, not Newton).

European thought post-Archimedes was in thrall to the twin curses of Plato's intellectual straightjackets and Aristotle's experiment free assertions. Plato elevated geometry to second rank amongst all pursuits - second only to philosophy (On the gates to Plato's academy were written "Let no one who is ignorant of geometry enter these doors"). Unfortunately, Plato's recognised only the axiomatic style of geometry and refused to countenance any other kind (such as drawing tangents to curves!). For 1800 years more, Europeans would be condemned to the sterile world of axiomatic geometry (where most interesting theorems had already been discovered by some smart Greek dude or the other) while the more interesting developments in algebra were taking place in Pataliputra (Ujjain?) and Baghdad.

As if Plato's stultifying intellectual hierarchies were not enough, along came his pupil Aristotle, the king of non-emperical studies, blithely asserting that heavier bodies fall faster than lighter bodies and purveying quite a lot of other untested rubbish.

The Europeans were in such awe of the Greeks that it became impossible to reject even flawed parts of the Greek cannon. Richard Feynman once asserted (in Greece, of course!) that the most important mathematical contribution from Europe was the solution of the cubic equation by Tartaglia in the fifteenth (or sixteenth?) century. This was the first time in 1500 years that post-Greco-Roman Europe had achieved a breakthrough that had eluded the ancients. And it probably gave Medieval European mathematicians the necessary self-confidence to pursue their later discoveries in the calculus.

Incidentally, the differential and integral calculus were discovered in Kerala in the 1400's by Madhava (written manuscripts exist in Malyalam) who almost certainly had the Fundamental Theorem of Calculus. There is good reason to believe that some Portugese missionaries from the early 1500's were exposed to this work and may have carried it back to Europe.

Ravi

Anonymous said...

Two typos in my previous post.

The relevant words should be spelled "canon" and "empirical".

Ravi

Dilip D'Souza said...

Ravi, I'm getting senile! Of course 1800 takes us to the end of the 16th Century. Thanks for that delightful exposition of the route that intellectual thought in Europe took. I like that idea that Archimedes, and not Newton, would have been the best person to understand Einstein. (Of course, Einstein's theory explains phenomena that Newton could not...).

On a different, but possibly related, note: what do you make of Socrates? Did the man make any original intellectual contribution apart from argumentation?

I look forward to more of this kind of thing from you.

Anonymous said...

I have to confess ignorance about Socrates, though I suppose his hemlock drinking ways have served as an inspiration to all intellectuals resisting tyranny (since the account of this incident rests primarily on Plato's word, one should probably treat it with some scepticism). I suppose too, that the Socratic method of discourse occupies a valuable place in pedagogy, though one suspects that he was hardly the first to practise it. In any event, he does seem to have inspired tremendous devotion in his students. We must also give him credit for apparently refraining from "scientific" pronouncements in sharp contrast to his intellectual descendants. Maybe Socrates gets away because as a first rate philosopher he took great care never to utter a single falsifiable statement!

By the way, Aristotle's babblings had much more immediate effects on the world - indeed the retardation of the march of science may not have been his worst legacy. As tutor to Alexander the Great he seems to have filled the latter with all sorts of incoherent prejudices against the Persians and their culture. In this, he was unable to rise above a rather vulgar nationalism - I suppose he was some kind of Henry Kissinger of his times (he taught at the Academy, Kissinger went to Harvard... etc.)- of his compatriots. Alexander went on to adopt many Persian customs and to respect their many contributions but not before he had vandalised much of their capital and their country. Had he not been indoctrinated from early childhood, perhaps a lest destructive monarch would have emerged.

To chime in, I too have always been sceptical about the parabolic mirrors story. For one thing, the mirrors would have to be extremely well-engineered to obtain a sharp focal point at a distance of several hundred metres (the mirrors would have had to be purely metallic since glass had not yet been invented by the Romans - still silver would have done, I suppose) and metallurgy had surely not advanced that far. In 10th class my classmate Samudra spent every afternoon patiently concentrating the suns rays through a magnifying glass on to his desk in order to engrave his name on it. As a labour of love it was impressive - it took him the better part of three months. Still, it shows that harnessing the sun for this purpose is no easy task, and a ship gently bobbing up and down in the sea by even six inches would be pretty much immune.

Ravi

Boskoe said...

Dilip,

Regarding the anecdote about Archimedes designing mirrors to burn enemy ships, a group of students from MIT set out to test whether there was an element of truth in it. The link is here http://web.mit.edu/2.009/www/lectures/10_ArchimedesResult.html

Enjoy!

Anonymous said...

I can't believe a Roman soldier, probably middle-aged, would kill a 75-year old man who's defenceless! That's just wrong! Plus, I read somewhere that the commander had given his soldiers strict orders that, in the event of you finding Archimedes, you SHOULD NOT kill him. Instead, simply bring him to me. I understand the soldier was angry, but the circumstances of the time were not favorable to kill Archimedes. I hope his commander let him have it once he found out.