100% is vertical. How can an incline be more than that?
There is a funicular in Switzerland claiming “a gradient of 106 per cent”. This figure widely quoted in many other sites that reference this railway.
The steepest road in the U.S. is, by one reference, Waipio Rd. in Honokaa, HI, at 45%. If you’ve ever driven on a steep road you’ll know that 45% doesn’t look that steep on a piece of paper but when you’re looking down over a steering wheel it is hellacious.
So what does 106% mean? It can’t possibly mean you are leaning over backwards.
I suspect that the “106%” means that, for every foot of horizontal movement, the funicular is going upwards (vertical movement) by 1.06 feet. That’s just a bit more than a 45 degree grade.
Adam Ondra’s climbing here is easy to express in the percentage format as a negative percentage. It’s just the boring Honnold vertical ascent up a wall that’s difficult to express because it’s infinity percent.
OK, on further research I see you are all correct. I think it’s a strange convention, but that is the convention nonetheless.
In analytic geometry, slope is defined as Δy/Δx, where a line at a 45° angle would have a slope of 1. But this is not expressed as a percentage in geometry.
If incline is simply the geometric slope expressed as a percentage, IMHO it’s not intuitive since you would expect 100% incline to be the steepest incline possible. As it is, the incline % is not linear (in fact it’s the tangent of the angle).
What is the reason for adopting a convention like this? Is there some practical aspect of engineering where this is a useful intuitive measure?
In surveying for civil structures (highways, landscaping, etc.), knowing the slope or incline, defined as [vertical] rise over [horizontal] run, quickly gives you a sense of how many feet up/down you move for every foot horizontal you move which is useful knowledge. It’s harder to figure that out if you are instead reading the angle in degrees. Example, a slope of 8% means you rise 8 feet for every 100 feet horizontal you travel. This is an incline of 4.57 degrees, but you’d have to calculate the tangent of that angle to find out the slope.
For the shallow angles usually encountered in civil engineering (steep highway hills may be around 10%, though some are a little bit steeper), the difference between your horizontal travel distance and your travel distance along the slope is negligible. The 106% slope of the funicular you mentioned is uncommonly steep, so the slope travel distance and horizontal travel distance are different by a large amount.
You’re thinking of “percent” as a fraction of a whole, which it is in many contexts. But “percent” can also mean literally “per hundred”. Off the top of my head, it means this in the context of geometry (106% slope), finance (125% tariff), and thermodynamics (200% efficiency).
Usually referred to as “grade”, at least in the US. Warning signs are usually posted approaching a steep descent - “6% grade next 2 miles”, for example. That is about as steep as I have seen on a limited access highway. We have some surface streets near my house with “10% grade” and “12% grade” warnings. Makes riding a bike challenging (at least for me and Mrs. Martian).
The steepest IBC (International Building Code) compliant staircase has a 7" rise and an 11" run = 63.6%.
In finance it makes sense for a rate of return to be greater than 100%.
In economics it makes sense for the increase in interest rates to be more than 100% (meaning, for example, that interest rates go from 0.5% to 1.25%).
In science, it makes sense for a measurement to be over 100% greater than the previous measurement.
In demographics it does not make sense to say that 106% of the population were born on planet Earth. When you are talking about a bounded quantity, you can’t have a portion of that quantity that exceeds 100% of that same quantity.
To my thinking, grade is a bounded quantity. You can have a horizontal grade, which is 0%. You can have a grade up to a vertical grade, which to my erroneous way of thinking is the maximum possible incline and seems like it would be 100%. However, to a road engineer vertical would be undefined, because it would involve division by zero. (As you approach vertical, the incline approaches infinity, but of course is never actually infinity.) The idea that you can express a finite quantity using a measure that approaches infinity and becomes undefined at its upper limit just bothers me. I guess it doesn’t bother civil engineers.
I’m not saying it’s wrong, I’m just saying it’s hard to me to think that way.
And that illustrates the source of the problem: there isn’t one meaning of “percent”, there’s two. There’s “fraction of a whole, expressed with a denominator of one hundred”, and there’s “relative rate, expressed as an output quantity per hundred input quantity”.
These are rates.
This is a fraction.
And this nicely shows you do know the difference. Your initial conception of grade is it’s a fraction of a whole, and your second is that engineers conceive of grade as a rate.
Actually the conveyance I linked to start this off doesn’t look like a funicular to me. I had the impression that a funicular had two cars, one going up and the other going down, and they are linked through cables underground so that the only energy needed to move them is what is needed to overcome the weight differential, inertia, and friction. That looks like just one car on one rail to me.
The picture shows only one car, but that site claims it’s a funicular. Wikipedia says one of the defining features of a funicular is two cars that offset each other’s weight:
So it seems there must be another rail car not shown in the picture you linked to.
Detroit airport’s mile-long terminal building has an express tram that operates similar to a funicular, except without the slope (OK fine, the slope is 0%…). There are two tram cars connected by a cable. When one is at the north end of the terminal, the other is at the south end; they pass each other at the midpoint on opposite sites of a boarding platform.
Pictures I can find of it online all seem to suggest it’s only a single track, so I suspect it’s not a funicular railway in the traditional sense, and is more like an inclined plane railway.
