DEVELOPMENT OP CLIFF PEOFILES. 
253 
cause of this curvature is as follows. If the rate of recession in the soft 
beds, as due to the protection of talus, were proportional in a simple ratio 
to the height above the bottom, the slope would be straight, but would 
gradually decrease its inclination as the cliff recedes. But, in fact, the law 
governing the rate of recession is more complex. The protection given to 
the lower beds increases downwards in a higher ratio than a simple one, 
being as the square of the distance below the base of C, or, perhaps, in a 
still higher ratio This arises from the fact that not only is the quantity of 
debris and soil greater in the lowest beds, but it is finer and more compact. 
Hence the rate of recession becomes inversely proportional to the square (?) 
of the distance below the base of C, and the curve becomes a segment of 
an hyperbola. 
The conclusions from the foregoing discussion are abundantly exempli- 
fied throughout the clifis and canons of the Plateau Country. Take, for 
instance, . the case of the Jurassic sandstone as one extreme. This rock is 
very hard and resistant and very homogeneous. It weathers with extreme 
slowness. The canons which are cut deeply into it, but not through it or 
deeply into the Trias below, are always narrow and have very bold and 
precipitous walls. The extreme is reached in the canons of the forks of the 
Virgen. These are merely narrow clefts in the rocks many hundreds of 
feet in depth. But as soon as the stream beds cut into the softer members 
of the middle and lower Trias the chasms at once widen out and then grow 
wider as the stream cuts lower. The talus at the base of the Jurassic es- 
carpments is always small and often entirely absent. It is, moreover, a 
general fact that the canons narrow up as they pass into hard rocks and 
widen out as they pass into soft rocks. 
In the foregoing discussion we have all the conditions necessary for 
understanding the cliff-work and sculpture of the Grand Canon, and we 
may now proceed to apply these principles to the peculiar profiles which 
the chasm presents in its several portions. The summit beds, consisting of 
arenaceous and cherty limestones, are of medium obduracy. They contain 
a large amount of silicious matter, and if blocks of it were submitted to the 
stone-cutter they would be pronounced excessively tough and hard ; but, 
owing to the presence of an abundant calcareous cement of soluble char- 
