> Then there are 'grades,' used on roads and probably elsewhere. For example, occasionally one sees signs warning of "6% grade next 10 miles" or something similar. What is grade?
>

I thought it was just the slope of the road (the vertical displacement divided by the horizontal displacement between two points) -- which would mean driving 100 feet forward up a 6% grade would make you ascend 6 feet.

I haven't ever looked this up, so it's all just my guess, but it seemed the most natural way to do it. If you have a contour map of a region, it's easy enough to measure the horizontal distance between two points using a ruler and the map's legend. If you knew the a road climbed at a constant angle between two points, you could use that information to determine the elevation difference pretty easily.

On the other hand, if you defined grade a different way, such as what percentage of a right angle your angle from the horizontal is, it would be more difficult to calculate elevation -- you'd need to take a sine or a cosine to convert from angles to horizontal or vertical displacement.