Ancient sources Edit

The thickness of the rope/cord/string used for the springs is discussed in Philon's Belopoeica. All translations below are from Marsden (1971: 113-119). The "M" refers to numbers in Marsden's edition of the said text.

"The diameter of the hole [D] is the unit of measure for all the individual parts in the engine." (Philon: Bel. M 53)

"Make the diameter of the thickness of the cord 1/4D minus [see below] 1/12D..." (Philon: Bel. M 54)

The description of the stretching process in Philon's Belopoeica also gives some indirect clues to the thickness of the spring cords:

"This beins so, let us now take the inconveniences which occur in the stringing. First, the stretching of the spring-cord (which is inclined to tangle) is a long job, because the strands are stretched one by one and, in each operation, the whole bundle must be threaded through the washers" (Philon: Bel. M 57)

Heron's Belopoeica contains a more thorough explanation of the final phase of the stretching process. Translation is from Marsden (1971: 37):

"When the latter [=spring cord] has been threaded through and the holes to receive it can hardly take more because they are full, it is necessary to push through the holes round iron bars, which are to be smooth and slender at the end. Knock them in with a mallet. When there seems to be sufficient space to take more spring-cord, you must continue putting it through. When, even so, it is received with difficulty, you must take an iron needle and thread the end of the cord through its eye. Then, pushing the needle itself through, haul the cord after it. When you think the holes are really full, cut off all but a little of the remaining spring-cord, if there is a lot left. But if there is only a little, never mind it, just wrap it round half of the spring." (Heron: Bel. M 108-109)

The descriptions of the stretching process are in stark contrast to the actual cord thickness given. As Stevenson (1995: 16-17) notes, Marsden (1971: 161-162) regarded "1/4D plus 1/12D" (=1/3D) as textual corruption and read "1/4D minus 1/12D" (1/6D).

Mathematical calculations Edit

If we assume that Philon's 1/3D is the correct figure, it's not possible to fit in even two ropes on each side of the washers. Let's take the Cheiroballistra as an example. It has the hole diameter (D) of 2 dactyls (d). Following Philon's (Bel. M 55) suggestion, the rope thickness should be 1/4*2 + 1/12*2 = 1/2 + 1/6 = 2/3 dactyls (~1.3cm).

Now let's see how the 2/3D thick rope would fit into the washers, if washer wall thickness and washer bar width are both 1/4d (~0.5cm). The area of one half of the washer minus the area occupied by half of the bar is roughly 0.53 square dactyls. One unstretcher rope occupies π * 1/3² ~ 0.349 square dactyls of space. So, it would only be possible to fit in ~1.518 ropes through each half of the washer. In practice this means only one rope would fit in each washer half. As discussed here, the thickness of the rope can't be significantly reduced during the pretensioning (stretching) process, so that does not help us one bit.

Practical experience Edit

Text analysis and mathematical calculations are only a part of the big picture. In order to end up with reasonably correct interpretations we need to test things in real life. As shown above mathematically, it's almost certain that the spring cord was less than 2/3D thick. However, for the sake of the argument, we can assume that the rope was so thick that 2 ropes could fit on both sides of the washer bar. In that case, two pairs of ropes (4 cords) would in theory go through the washers. If we used such a rope, we'd end up with three practical problems:

  • How do we fasten the beginning of the rope without consuming the precious space inside the washer? For thin rope we can easily use a three-ply eye splice loop attached to the washer bar; if we did the same for the theoretical, almost 2/3D thick rope, we could only fit in 3 cords instead of 4. In other words, we'd lose 25% of the power.
  • Where do we attach the end of the spring cord? We can lash a thin cord to another cord without loosing much power. With near 2/3D thick rope, we can't do this without losing yet another 25% of the potential power output.
  • An incredibly large and powerful stretcher and clip would be needed to pretension the rope. In the case of the cheiroballistra the pi-brackets probably couldn't take the stress.

Practical experience has shown that 1/6D is probably too much, even for a small engine such as the cheiroballistra, where the 1/6D cord thickness turns out to be 1/3 dactyls (~6.66mm). I have not experimented with cord that thick, having used 4mm nylon three-ply exclusively so far. With 4mm cord one can fit six cords next to each other on the first layer of cords. The subsequent layers can take five, four and one or two cords, respectively. Even with the 4mm cord there are several issues:

  • There's quite a lot of empty space around the cords, especially around the last layer.
  • Small irregularities of the washer hole can significantly reduce the number of cords that can be inserted.
  • Clipping the cord is rather problematic, as the pull of the cord is considerable. The simplistic clother's peg style clip apparently described by Heron (Bel. M 108; see Marsden 1971: 37, 58) definitely does not work. Modern locking pliers are adequate for high pretension and 4mm cord, though, so clipping cord this thick is definitely doable.

Trying to increase the spring cord diameter to 1/6D (=6.66mm) would increase the pull of the cord ~2.77 times, which would probably make clipping almost impossible. Moreover, any clip will only be able to grip the surface of the cord, and the pull increases faster (to the power of two) than the surface area of the cord (linearly), given the same amount of pretension. Thus the cords are more likely to get damaged by slipping through the clip if they're thick than if they're thin. In extreme cases clipping would be impossible.

Different kinds of problems arise if we try to use cord that's too thin. For example, each cheiroballistra spring needs ~10m of 4mm spring cord. If we halved the cord thickness to 2mm, the number of cords required would quadruple (2²), and we'd need 4*8 = 32m of cord. All of this 32 meters would have to be passed through the washers and the winch and pretensioned without the cord becoming a tangled mess at any point. As discussed here, it takes 2-3 hours to carefully arm the cheiroballistra with 4mm cord, not counting any preparations. It would thus take 8-12 hours to arm it with 2mm cord, even if we disregard the additional time required to haul all the 32 meters of cord through the washers. The hauling time could be reduced if several shorter pieces were attached together as more cord was needed. If the cord was thin enough, a few knots or splices here and there would not matter much, whereas in thicker cord a significant percentage of the spring's cross-sectional area would be consumed. Joining smaller lengths of cords would not reduce the time spent winching, measuring the pitch and clipping the cord. At this point it should be noted that if the pitch can be gauged accurately by ear, the process would be significantly faster than when having to use a modern tuner for measuring the frequency.

Based on practical experience the optimal cord size seems much less than 1/6D. That number also clearly contradicts the accounts of Philon (Bel. M 57) and Heron (Bel. M 108-109), which indicate the use of a fairly thin cord. I suspect that in smaller machines the cord thickness was <2mm, primarily because that is about as thick as one can make from sinew by hand as two-ply in one go. In larger machines real rope composed of several simple cords was probably used. In that case clipping would be more tricky, but less of a problem than doing thousands of passes through the washers and the winch.

Note that the above problems somewhat specific to standard ballistas where the cord has to be passed through the washers. In Philon's wedge engine (Philon: Bel. M 59-68) thin cord could be used more easily, as it can be wound to the frame directly from a spool. That said, the other benefits of thin thread are somewhat lost in that type of machine.