an autodidact meets a dilettante…

Rise above yourself and grasp the world.

Archimedes (attributed: inscribed on the Fields Medal)

One of Archimedes’ most spectacular inventions, the gravity-defying spiral-in-a-cylinder, or screw – still effective

Canto: So we spent some time at the Waikato museum in Hamilton, braving the school holiday crowd to view an exhibition celebrating the work of Archimedes (c 287-212 BCE) and his fellow mathematikos, and noting how it inspired the likes of Leonardo some 1800 years later. So let’s talk about their breakthroughs and about why there was such a gap between their clever contrivances and the maths on which they were based, and the scientific revolutionaries of the sixteenth and seventeenth centuries.

Jacinta: These are intriguing and vital questions. Many modern scientists have been dismissive of the science of the ancient Greeks because they think of Aristotle as representative. I think it was Lawrence Krauss I heard complaining of Aristotle’s belief that women had less teeth than men – apparently he never thought to count them! But the fact is that, though Aristotle is sometimes known as the father of empiricism, he probably doesn’t deserve that title except in respect of ethics, and politics, which he based on what actually works for societies and city-states, which is why he collected and analysed their constitutions. The mathematikos, on the other hand, eschewed ethical issues in favour of mathematics – geometry in particular (think Euclid). And, especially in the work of Archimedes, they enjoyed phenomenal success in many practical areas.

Canto: Especially warfare apparently. It seems Archimedes in particular was called on more than once to defend his city, Syracuse, with war machines. In the blurb to the exhibition, they mention ‘torsion ballistae’. Can you please explain?

Jacinta: Well, I’ll tell you about the torsion siege engine. It replaced the earlier tension siege engine, possibly invented in Syracuse in the time of Dionysius the Elder (c 432-367 BCE) – so the engineering of weapons of war was already a big thing at the time. It was basically a massive catapult. The first torsion device of this kind is generally dated to the time of Philip II of Macedon, Alexander’s dad, circa 340 BCE. The first extant evidence of its use comes from a list of items in the arsenal of the Acropolis in Athens dating to 338-326 BCE. So what is torsion? It’s the energy created by winding something up, like a spring. In earlier times, human hair, horsehair and animal sinews were used for this purpose.

Canto: So plats give you energy?

Jacinta: Torsion basically means twisting. The Greeks apparently used specially cured sinew combined with human or animal hair to create a ‘torsion bundle’ – we don’t know what the exact recipe was – which was fixed to a wooden frame and could be twisted and released regularly via levers without breaking. But the key development was the mathematics of these devices. This military website describes:

The critical dimension was the diameter of the sinew “spring” or torsion bundle. For a bolt shooter, the ideal diameter was one-ninth the length of the bolt. For a rock thrower the ratio was more complex; the diameter (d) of the bundle in dactyls (about 3/4 inch) should equal 1.1 times the cube root of 100 times the mass of the ball (m) in minas (about a pound). Saddled with a numerical notation system even more awkward than Roman numerals, the Greeks developed sophisticated geometric methods to compute cube roots.

Canto: So how were these maths – these geometric methods – derived. Euclid was the great geometer of the time, wasn’t he?

Jacinta: Actually, though the exact time-frame of Euclid’s life isn’t known, his Elements came out after this invention, but before the work of Archimedes. Clearly it must’ve been drawn from earlier mathematikos, such as Eudoxus, who worked out, via an early version of integral calculus, that areas of circles relate to squares of their radii, and volumes of spheres relate to the cubes of their radii, and various other relations between volumes and dimensions of pyramids, cylinders, cones etc, which obviously had practical applications as described above.

Canto: Okay, so tell us about Archimedes’ particular contributions, and about why the great work of the mathematikos was apparently discontinued after Archimedes. Considering that the Roman Empire didn’t become christianised until some 500 years after Archimedes’ death, we can’t really blame the Christians – can we?

Jacinta: Well, I mentioned that early version of integral calculus. It was called the method of exhaustion, a kind of geometric calculus which Archimedes took further than anyone before him, both in theoretical terms and via practical applications. Now I’m far from being a mathematician, but I’ve come to appreciate the essentiality of maths in understanding our universe – so much so that I perhaps regret my lack of mathematical expertise more than I regret anything else in my old life. This is by way of saying that I won’t try to explain Archimedes’ maths – but an understanding of maths is essential to understanding the magnitude of his achievement.

Canto: Okay, so what about his inventions?

Jacinta: Well the key is the application of complex and what might have seemed pointlessly abstract maths about the relations of ‘perfect’ shapes such as spheres, cones and cylinders to real world problems and their solutions. The lever is a good example. Archimedes didn’t invent levers but he was clearly fascinated by them. And it shouldn’t take long to realise that they have immense practical applications. Doors are levers, as are nail clippers, nutcrackers and see-saws. Archimedes wrote what we now call a treatise, On the equilibrium of planes, to explain the maths behind them. But the best illustration of Archimedes’ combination of theory and practice is probably what is known as Archimedes’ principle, which essentially launched the field of fluid dynamics, or fluid mechanics:

the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces and acts in the upward direction at the center of mass of the displaced fluid.

It comes from another treatise of his, On floating bodies. Now let me see if I can explain this. Take an object, any object. The downward force exerted on it is its weight. Immerse it in water. It will float or it will sink, and even if it floats it’ll be partially submersed. The principle doesn’t apply to objects that sink, which will have a density of anything over and above a certain level which permits it to float – I think.

Canto: But that principle, though it comes from a treatise about floating bodies, doesn’t distinguish between floating and sinking. It says ‘fully or partially submerged’…

Jacinta: But an object can be fully submerged and still float. To sink means to continue in a downward direction.

Canto: I’ve found that an object that floats – I’m thinking of water as the fluid, and perhaps I shouldn’t – always seems to have a certain proportion above the water level. Think of icebergs, and human bodies. But I think I get it – the force that keeps you up and floating will be equal to the weight of the water your body displaces… So if I was ten kilos heavier, I would still float but the upward force acting on me would be greater, but not by ten kilos, rather by the larger volume of water my larger body displaces measured in kilos, or by some measure of force…

Jacinta: I don’t think that’s wrong, but I’m not sure if it’s right. The problem for me is that the principle as stated doesn’t specify a floating body, only a body immersed – partially or fully in a fluid. Think of a stone dropped in water. It sinks. To the bottom.

Canto: And if the fluid is bottomless will it just keep on sinking? It’s as if there’s no upward force acting on it at all, or very little. I’m imagining a bottomless column of still water here, not an ocean with its currents…

Jacinta: Ha, I was thinking of a bathtub, but with a bottomless well, it will depend on the density of the stone. I think at some point it’ll slow down and be suspended. I’m sure water pressure will play a role, and density – of the water. And density is somehow related to pressure, and I’m getting lost…

Canto: We may need to do a Khan academy course. But getting back to Archimedes and the mathematikos, why was so little of their work built upon, until Galileo and others became inspired so many many centuries later?

Jacinta: That’s possibly too long a story to go into here, not that I’m much equipped to tell it. It no doubt relates to the gradual decline of the increasingly dispersed Greek culture of the Hellenistic and post-Hellenistic era. I wouldn’t want to say Christianity was a major cause but it certainly didn’t help. By the time the Roman Empire became Christianised, the culture that created figures like Archimedes had long passed. Roman culture was a lot more militaristic and less speculative. Blue sky research wasn’t in vogue. Of course, why all this happened I wouldn’t venture to say without many years of research into the cultural changes then occurring. But the slowness of the scientific recovery, that I would attribute to Christianity, and later to the conservative turn in Islam that still prevents original science from being practiced in those countries where it holds sway.