Naismith's (1892) rule:
Hiking time = Allow 1 hour for every 3 miles (5 km) forward, plus 1 hour for every 2000 feet (600 metres) of ascent.
Aitken (1977) - Langmuir (1984) corrections for descent slopes:
Aitken: 5 km/h can be maintained on paths tracks and roads, while this is reduced to 4 km/h on all other grounds.
Langmuir: minus 10 minutes per 300 m descent for slopes between 5 and 12 degrees; plus 10 minutes per 300m descent for slopes greater than 12 degrees.
However if you assume the initial speed of 5 km / h, Langmuir correction gives enormously high hiking speed of 12.2 km / h at -12 deg slope, whereas the maximum walking speed for young humans is about 9 km / h.
But if you take 4 km / h as basal velocity, you obtain 7.58 km / h at -12 deg, what is much better result. So we would have
4 km / h (or 5 for 5 km / h) between 0 and -5 deg,
4.97-7.58 km / h from -5 to -12deg and
2.72 km / h and less below -12 deg.
What exactly Langmuir has written in his book ?
An example of another rule is Tobler's hiking function, but it contains one maximum of 6 km / h at -2.86 deg (with 5 km / h at 0 deg) instead of Langmuir's dramatic thresholds, especially that one at -12 deg. According to some papers, a threshold in this area occurs.