Everything in this chapter is built from two measurements and the ways of writing their ratio. Get the four names straight now and nothing later will surprise you.

The rise, written $\Delta h$ — read aloud delta h, where delta simply means “the change in” — is the difference between two elevations, in metres, measured straight down the plumb line. It is positive climbing and negative falling, and a fall is just a rise wearing a minus sign. The run, $L$, is the horizontal distance between the same two points, also in metres, measured with the tape held level. Not along the ground. That third length — the slope distance a tape reads lying on the surface — is longer than the run and belongs to a different calculation entirely.

Divide one by the other and you have the percent grade: $G = \dfrac{100\,\Delta h}{L}$, read aloud G equals one hundred delta-h over L. $G$ carries no unit of its own — both metres cancel — and the hundred is there only to move the decimal point into the form a drawing prints. A 2 % grade falls two metres in every hundred metres of level run.

The fourth form is the slope ratio, written n:1, and it is the one that catches people. The convention is horizontal first, vertical second: a 3:1 batter lies three metres out for every metre it climbs. So a bigger first number means a flatter slope, which is the reverse of how a percent behaves. Hold the three together: 3:1 is 33 %, and 33 % is about 18.4°. They are the same slope said three ways, and any two of them look nothing alike.

One more piece of the language. A station or chainage is a position along the centreline, written with a plus sign — ch 0+150 is 150 m along. In the United States a station is 100 ft and 12+50 means 1250 ft; in metric practice the plus sign survives but the count is metres. Either way it is a position, not a distance, and the distance between two of them is what becomes your run.

Units point the way in this lesson. A quantity in metres is a length — rise or run. A quantity with a percent sign is a grade. A quantity with a colon is a ratio. A quantity in degrees is an angle. Nothing here needs arithmetic; it needs reading.