Archimedes, The Sand-Reckoner


The greatest of all ancient scientists, and the only one fit to stand shoulder-to-shoulder with Newton, Galileo, and Einstein, was born in Syracuse, Sicily, around 287 B.C.  He happened to be the son of an astronomer named Pheidias; and perhaps this fact played some role in the selection of his career.  He moved to Alexandria after completing his schooling in Greek Sicily, for Alexandria was in those days the Mediterranean world’s capital of learning.

Like Einstein and Newton, he was so absorbed in mathematical and scientific inquiry that he would neglect his health and hygiene.  His friends and colleagues nicknamed him the “sand-reckoner” for his habit of pondering mathematical problems by inscribing them on the ground.  We are also told that he deliberately inserted false theorems into his works (especially in his work The Sphere and the Cylinder), partly to tease his friends, and partly also to identify thieves who would steal his ideas.

He came within a hair of inventing integral calculus (some say he did in fact do this, using his “method of exhaustion”) and algebraic analysis (see his famous Cattle Problem).

Europe would not catch up to him until Newton himself definitively produced the mathematics of calculus, and that would be over 1600 years later.  His treatises remain masterful examples of mathematical art; and we are left breathless in wonder at how one mind could have achieved so much in so little time.

Of his published output, ten works have survived.  The titles alone say it all:

The Method

A Collection of Lemmas

The Measurement of the Circle

Quadrature of the Parabola

On Spirals

The Sphere and the Cylinder

On Conoids and Spheroids

The Sand-Reckoner

On Plane Equilibriums

On Floating Bodies

Readers with a mathematical bent, and interested in the history of science, owe it to themselves to give these treatises a look.

No account of Archimedes would be complete without relating the famous story of his discovery of the law of weight displacement in liquids.  King Hieron of Sycuse, we are told, had hired an artisan to fashion a gold crown for him.  To this end the king supplied him with some gold.  The finished crown weighed the same as the allotment of gold provided, but the king had suspicions that the crown may have been alloyed with some cheaper base metal.

Archimedes was brought in to consult on the problem.  How could a way be found of learning the truth, without destroying the crown?

The great scientist finally noticed one day, when bathing, that his body seemed to displace water in relation to the depth of his immersion.  His body also seemed to weigh less the deeper it was pushed into water.  From this, he was able to deduce that the weight of water displaced by a floating body is equal to the weight lost by the floating body.  And from this, he concluded that a fully submerged body must displace water in relation to the body’s volume.

Whether this discovery actually prompted the scientist to run through the street shouting Eureka, Eureka, I will permit the skeptical reader, alert to the amusing apocrypha of history, to decide for himself.

But how would this discovery help him in solving the problem of the crown?  A base metal or silver would weigh much less than gold.  Therefore a given weight of a base metal would have more volume than the same weight of gold.  And it would also, therefore, displace more water.  From doing experiments, he verified that Hieron’s crown displaced more water than an equal weight of gold.  The only possible explanation could be that the crown had been alloyed with a cheaper base metal or silver.  Archimedes was even able to calculate just how much gold had been stolen.

The king, in other words, had been short-changed.  Archimedes became famous for discovering how to measure specific gravity.  The artisan was executed.


He was also an eminently practical scientist, in an age when theoretical speculation too often took precedence over real-world application.  He constructed a functional planetarium.  He worked out the laws of pulleys and levers so completely that no improvement was made to his work until 1586.  “Give me a place to stand,” he is supposed to have said, “And I will move the world.”

It is a compelling comment on the power of the mind over the material world.

Like Leonardo da Vinci many centuries later, he was fascinated by weapons and machines of war.  He did these tasks not out of a love for war, but simply out of his interest in solving problems of mechanics and physics.  He also made practical advances in agriculture and irrigation that are used to this day:  the so-called “Screw of Archimedes” is a simple and practical way of putting to use his principles of spirals to elevate water.

When the Romans undertook the siege of Syracuse as part of their expansion in the Italic peninsula, the old scientist (then seventy-five) was compelled to assist in the defense of the city.  Archimedes’s machines–cranes, pulleys, catapults, and giant hooks–held off the Romans for a great deal of time.  The historian Polybius (I.100) so commented:

Such a great and marvelous thing does the genius of one man show itself to be when properly applied.  The Romans, strong both by land and sea, had every hope of capturing the town at once if one old man of Syracuse were removed; but as long as he was present, they did not venture to attack.

But time did tell in this matter.  After eight months of siege, the Roman commander, Marcellus, was able to take the starving citadel.  He was a great general and a good man in his own right, and he gave strict instructions that Archimedes was not to be harmed.  Yet such are the passions of war and conquest that they often overwhelm the intentions of decent men.

The old scientist was found–we are told–by a Roman soldier unaware of his significance.  The soldier ordered him out of his house, but Archimedes asked him to wait a moment until he could solve a problem he was working on.  In the dispute that followed, he was slain.

Marcellus ordered compensation for his relatives, and had a grave marker erected in his memory.  On the huge stone’s surface was inscribed a sphere within a circle, in accordance with the old man’s wishes.  He believed that the discovery of the formulas for the volumes of these things was his greatest achievement.

With him, classical science went to the grave.  Scientific inquiry would not achieve such heights for another 1600 years.


Read More:  The Genius of Joseph-Louis Lagrange


6 thoughts on “Archimedes, The Sand-Reckoner

  1. Thank you for another great, well written and researched article.

    I have been reading your blog for some time, and I believed it was time that I stop being lazy and write a comment, thanking you on the high quality of your posts, for which you greatly deserve.

    I love to read the old classics, but due to my chronic health and pain (and just getting old) I have been unable to read as much as I would wish. Interestingly enough, this has been the thing that I miss the most.

    So I would like to say well done, not just for this post, but for many more that have come before – and thank you for opening up new ideas that I would be unable to discover myself due to my disability.

    I would also like to encourage you to continue your good work, even though readers don’t comment, they are reading, and I strongly believe, you are making a great difference.

    Your blog stands in quality as one of a kind.

    So I would like to wish you a happy Christmas and may you continue your great work in the new year.


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