As Aequus as Aqua

Earth curve

Or, translated from the Latin: “As Level as Water.” As we explored ancient layout tools at length in “From Truths to Tools,” it became quite clear that the artisans of antiquity were no dummies. For example, we see from their tools and works that they understood that there was a difference between the curved “level” of a horizontal line and the straight “level” of a sight line. In fact, when they used the term “horizontal” to name the latter they were alluding to “horos,” the horizon, the boundary between water and sky.

How did they know that the earth they stood on was a sphere? Two things for starters, according to source documents: They observed the arc-line shadow of the earth falling on the moon during a lunar eclipse, and they watched ships disappearing on the horizon from the hull to the top of the mast (as opposed to the ship simply getting smaller and smaller). Why is this so important? Try building an aqueduct so it works properly or digging a tunnel accurately through a mountain without accounting for this difference.


If the trough of the aqueduct were constructed to a sight (or laser!) line level, the water would flow toward the center because the center, relative to the earth’s surface, is downhill from either end. Another problem that could arise if the support columns were constructed to meet the trough at right angles, is that the columns would only be plumb in one location. They would all be parallel, but that doesn’t make them right! (Literally: the forces on the un-plumb columns would have some amount of shear in them, leading eventually to distortion and ultimately failure.)


In tunnel work, the opposite problem arises: if they relied solely on horizontal level as the digging progressed from start points established by sight lines shot around the mountain from surrounding benchmarks, the tunnel would not exit at the predicted opposing point. Digging from either end, one tunnel would travel above the other and they would never meet. Note that the gradient drop of the earth’s curve is about 8′ per mile – and it’s not an additive (linear) increase, but exponential to infinity. To grasp this intuitively, picture the earth constantly curving away from the sight line. Eventually, at a point just past a quarter of the way around the sphere, a line dropped down square from the sight line would never reach the earth’s surface.

The more George and I immersed ourselves in research for this book, the more we gleaned about the tools and works of the artisans of antiquity and the smarter they started looking to us. The corollary was the dumber the guys in the mirror looking back at us each morning started to look! Obviously, not only is there is still so much more to learn, there is so much more to relearn!

— Jim Tolpin,

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10 Responses to As Aequus as Aqua

  1. nrhiller says:

    Fascinating post. I especially appreciate your observation regarding common assumptions about intelligence increasing with our species’ advancing years.

    • hgordon4 says:

      It’s an interesting point to ponder. I’d argue that the horizon of our knowledge, as a species, is definitely advancing – frighteningly rapidly in some respects. But the level of understanding of what’s going on around him, for the average person – especially in the industrialized world – has regressed rather badly. I read somewhere recently (sounds like a Mike Rowe thing maybe?) that the greatest divide in our society is between those that know how to do things with their hands and those that utterly do not.

  2. cdcook82 says:

    “Note that the gradient drop of the earth’s curve is about 8′ per mile …” Should that be 8 inches?

  3. SSteve says:

    I’m a computer programmer, not a civil engineer. But I worked in a physics lab for eight years so I got scale and units drilled into me. When I saw an aqueduct where the curvature of the earth was that significant, I thought “Wow. That’s a long aqueduct.” My obvious next thought was “How long?”. So I pulled the image into Pixelmator and did some measuring.

    Length of chord described by arc: 1206 pixels
    Height of arc: 74 pixels
    Calculated ( radius of circle described by arc: 2494 pixels
    Equatorial radius of earth: 6378 km
    Image scale: 2.557 km/pixel
    Aqueduct length: 833 pixels = 2130 km = 1323 miles

    Wow. That’s a long aqueduct.

  4. jtolpin says:

    While exaggerated, its worth noting that the Roman Emperor Claudius had a tunnel built in 41 AD that ran 3.5 miles. Aqueducts ran much further of course.

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