A Beginner’s Guide to Drucraft #12: Euler’s Limit

Sigls can only be created from essentia.

The first of the Five Limits, Euler’s (pronounced “oilers”) Limit is also the most obviously important one.  Sigls can only be created from essentia, and this essentia can only be found in sufficient quantities in Wells.  No Wells means no essentia means no sigls means no drucraft.

Euler’s Limit is named after Leonhard Euler (1707-1783) though it should be noted that Euler himself never tried to attach his name to the principle – he viewed himself as primarily a theoretician, and gave credit for the details of the discovery to Isaac Newton’s work in the early 1700s.  Euler’s name stuck, however, partly due to his work on the Five Limits as a whole, and the name “Euler” is now used daily by drucrafters who have no idea who Euler even was, to the great irritation of mathematicians.

Euler’s Limit Explained

When a sigl is created, it takes the form of a small object resembling a gemstone, measuring roughly 2mm to 16mm in diameter.  The essentia used to create the sigl takes the solid form known as aurum.

A sigl can be thought of as something like an electric circuit.  Personal essentia (the electric charge) is fed into it;  this flows through the channels of the sigl (the wires of the circuit) and produces a spell (the circuit’s output).  The function of these channels is to “conduct” the essentia.  Sigls work because aurum is conductive to essentia – if you try to use other materials for the “wires”, the essentia won’t flow through them.  Thus, to create a sigl, you need aurum.

(IMPORTANT NOTE:  this analogy falls apart as soon as you look at it too closely.  Obviously, essentia does flow through things that aren’t aurum – it flows through everything.  But the electric circuit model is quite a useful one for describing the way sigls actually work, so it’s a good place to start.  In pre-electronic times, drucraft teachers typically used the analogy of water in a hydraulic system, instead.)

This creates a significant problem:  while essentia is abundant, aurum is not.  In fact, ounce for ounce, sigl-grade aurum is one of the rarest and most expensive substances on Earth.  As such, there has always been an enormous incentive for drucrafters to find an alternative.  Developing an “aurum substitute” has been one of the three Holy Grails of drucraft research for centuries:  any company, government or House that could develop one, and produce it consistently and cost-effectively, would become unimaginably wealthy almost overnight.

However, so far this has not happened, and there is some reason to believe that it’s never going to happen.  Newton’s research theorised that the unique properties of aurum derived specifically from its point of origin, and he even produced a mathematical proof to that effect.  The proof has been questioned, resting as it does on certain unproven assumptions, but the general consensus among drucraft researchers today is that aurum works the way it does because it’s comprised of solidified essentia.  In other words, it’s impossible to find a substitute for aurum that doesn’t require essentia, because it’s the fact that it’s made out of essentia that makes it behave that way in the first place.

This means that sigl production, and thus all drucraft, is fundamentally bottlenecked by the global supply of Wells.  This, along with the Blood Limit, shapes the drucraft economy.


More so than any of the others, Euler’s Limit has stubbornly resisted any attempts to bypass it.  Over the centuries, drucraft researchers have tested a truly astronomical number of substances in attempts to find one that would serve as a substitute for aurum.  They’ve tried every naturally-occurring mineral on Earth (as well as more than a few non-natural ones), a dizzying variety of organic compounds, every element in the Periodic Table, vacuums, and even matter in exotic states.  All have failed.

The best that drucrafters have managed to do is to reduce the volume of aurum in a sigl.  While it seems to be impossible to use no aurum, it does seem possible to use less aurum.  This was the inspiration behind the invention of threaded sigls, which replace nonessential parts of a sigl with pockets of vacuum or gas.  Threaded sigls proved to be significantly more cost-effective than solid ones, and their use quickly spread.

These savings, however, came with a tradeoff.  It turns out that while much of the aurum in a solid sigl isn’t necessary, it does a lot to stabilise it.  Aurum in its physical state naturally sublimates, turning back into free essentia, and shaping a sigl with a threaded design instead of a solid one accelerates this process – often massively so.  While solid sigls can last for hundreds or even thousands of years, threaded sigls have lifespans that are only a fraction of that – heavily threaded sigls typically last two or three years at the most.  Their tendency to fail with time or under stress has made many drucrafters hate threaded sigls with a fiery passion, though this hasn’t stopped them from becoming the industry standard.  A solid sigl will last for a drucrafter’s entire life;  a threaded sigl will break after a few years, forcing them to go back to the manufacturer and buy another one.  Most companies see this as an advantage rather than a problem . . . the fact that they can produce threaded sigls at a lower cost as well is just a bonus.

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9 Responses to A Beginner’s Guide to Drucraft #12: Euler’s Limit

  1. Celia says:

    Oops. I was pronouncing it “you-lers.” Good to know!

  2. Nick says:

    Blush! Me too, after all these years I’ve been getting it wrong too!
    Thanks for another update on the world building, how is book #3 going? Last time that we heard you had struggled to complete chapter 3 to your satisfaction, which might impact your aim on finish the whole book by mid-year…

  3. Bill says:

    Glad to hear that you are making good progress in your writing!

    As threaded Sigls contain less essentia does the “Less personal essentia” mean that they are more lightly bonded to their owner and (perhaps) could be used by a wider range of relatives? This would allow a Finders Stone (for example) to be used by the extended family.

    Whoops! I suppose you may have been planning to be covered in the Blood Limit section…

    • Benedict says:

      No, threaded sigls contain roughly the same amount of personal essentia – this is the kernel, which is pretty much the same in both solid and threaded sigls. It’s the outer layers of a threaded sigl that are different.

      Threaded sigls don’t outperform solid ones in any significant way. Their only advantage is their cost.

  4. Nick says:

    It occurs to me that a lot of Essentia must be tied up in Sigls that no longer work as they were either faulty (Stephen has a fair number of these) or that their linked owner is now dead. As a sigl takes a lot of essentia to make (sometimes a whole well’s worth).
    Can the essentia in these sigls be released to (potentially) make new sigls? It would certainly give Stephen a lot more to experiment with…

    • Benedict says:

      Yes, that’s absolutely something that happens, and it’s a significant part of the drucraft economy. However, recycling sigls isn’t anywhere near as profitable as making new ones, because most of the essentia ends up getting wasted in the process. That’s really a whole article on its own, though.

  5. Rob says:

    So, since aurum naturally sublimates back into essentia over time, is it possible for aurum to be “recycled” back into essentia to be used in producing a new sigil? In other words, could a crafter take a pile of old, failed, threaded sigils, and use them as source material for a new sigil?

    • Benedict says:

      Sort of. It’s relatively easy to ‘recycle’ old sigls back into free essentia, but it’s quite hard to turn that essentia into a new sigl. Partly this is because the essentia tends to dissipate, and partly because if you want a high-quality sigl, you really want for as much of the essentia as possible to come from the same Well – ‘patchwork’ sigls are much weaker.

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