Archimedes & Leverage. a first note

Archimedes did not invent the technology of the lever, but deployed a little geometric reasoning to exploit its basic physical principal. Plutarch relays Archimedes’ use of the principal of mechanical advantage, by way of simple block-and-tackle pulley, to lever a ship by hand.

‘…Archimedes, who was a kinsman and friend of King Hiero, wrote to him that with any given force it was possible to move any given weight; and emboldened, as we are told, by the strength of his demonstration, he declared that, if there were another world, and he could go to it, he could move this. Hiero was astonished, and begged him to put his proposition into execution, and show him some great weight moved by a slight force. Archimedes therefore fixed upon a three-masted merchantman of the royal fleet, which had been dragged ashore by the great labours of many men, and after putting on board many passengers and the customary freight, he seated himself at a distance from her, and without any great effort, but quietly setting in motion with his hand a system of compound pulleys, drew her towards him smoothly and evenly, as though she were gliding through the water.’ [1]

If later, an economic historian like A.P. Usher turns the latter-half of his craft to write a History of Mechanical Inventions, and in the process observes that Archimedes achieves his great material feat on the back of the block-and-tackle pulley, it’s because such instrument is a subclass of a class of instrument known to financial economics and mechanics alike –namely, the instrument of a ‘lever’.

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To lever is to leverage. And leverage, in its utmost generality, is that activity which –in a thoroughly non-theological sense– miraculously seizes while yet preserving an input power to manage force against movement, to therein realize an excess of output force incommensurate with its input. In mechanics, as our careful reader will already detect in Plutarch’s version, and no doubt understandably so, we see all emphasis fall to the spatiality of leverage, with little more than an implicit nod to the productive role of positive differences whose spatial mediation requires a certain metrics of oscillation, and is therefore also and equally so temporal in kind.

To then commence our inquiry into leverage, we will naively implore Archimedes: ‘to lift a heavy ship by mortal hand? – Archimedes, please tell us, how this is possible?’

His curt reply will be, ‘through leverage.’

Or when feeling more loquacious, he may even elaborate, ‘by way of agent employing an otherwise inert instrument to exploit a set of positive differences between spatio-temporal properties.’

Or more fully still, that ‘when a quasi-causal operator arranges differences in force at different distances around a fulcrum, a certain timing is arranged in space –or rather a spatial distribution begins to ‘take up’ difference, to make up difference in time; the order of this distribution is the production of differences in temporalities, different temporalities are differentiated. This is, in short, the transformation of an inert object of the lever into a powerful technology capable of pivoting-on, and exploiting, but simultaneously (and paradoxically so) also producing such differences.

For indeed, we will now add, and by repeating our definition of leverage in its ‘utmost generality’: when positive differences are arranged at a sensitive point in an oscillation, at a critical value, and with the correct quantity and quality of inducement, the otherwise inert fulcrum spins on its singularity, and transforms into a dynamical instrument capable of producing an amplification of output power in excess of its original input. How far removed is this dynamic of leverage from the dynamic of leverage deployed in finance. As we will see, not very. It’s all a matter of knowing the difference between artificial leverage –sedentary, cardinal, convergent, and non-additive by nature– and natural leverage –which is nomadic, ordinal, divergent, and non-additive by nature.

So from here we commence our inaugural examination of financial leverage, and as we begin to move towards the proposition of infinite leverage. What do we have? We have a singularity combined with a set of affects, i.e. capacities to affect and in turn to be affected; there is the arrangement and exploitation of positive differences in space-time through intervention by a quasi-causal operator; and the dynamical use of an inert instrument to affect nonadditive outputs to matter and energy. These will suffice as our illustratives –our material illustratives for the miraculous power of leverage.


FOOTNOTE.

[1] This passage is worth reading in full:

‘For the art of mechanics, now so celebrated and admired, was first originated by Eudoxus and Archytas, who embellished geometry with its subtleties, and gave to problems incapable of proof by word and diagram, a support derived from mechanical illustrations that were patent to the senses. For instance, in solving the problem of finding two mean proportional lines, a necessary requisite for many geometrical figures, both mathematicians had recourse to mechanical arrangements, adapting to their purposes certain intermediate portions of curved lines and sections. But Plato was incensed at this, and inveighed against them as corrupters and destroyers of the pure excellence of geometry, which thus turned her back upon the incorporeal things of abstract thought and descended to the things of sense, making use, moreover, of objects which required much mean and manual labour. For this reason mechanics was made entirely distinct from geometry, and being for a long time ignored by philosophers, came to be regarded as one of the military arts.

And yet even Archimedes, who was a kinsman and friend of King Hiero, wrote to him that with any given force it was possible to move any given weight; and emboldened, as we are told, by the strength of his demonstration, he declared that, if there were another world, and he could go to it, he could move this. Hiero was astonished, and begged him to put his proposition into execution, and show him some great weight moved by a slight force. Archimedes therefore fixed upon a three-masted merchantman of the royal fleet, which had been dragged ashore by the great labours of many men, and after putting on board many passengers and the customary freight, he seated himself at a distance from her, and without any great effort, but quietly setting in motion with his hand a system of compound pulleys, drew her towards him smoothly and evenly, as though she were gliding through the water.’

1 thought on “Archimedes & Leverage. a first note”

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