MF: A 
332 On the Mammoth Cave of Kentucky. 
crease with the time z, we obtain the series 1, 17, 2¢?, 3x°, ete., 
wherein appears the leading numerical factor concerned in the 
process of differentiation: or if the unit of operation be negative, 
the series becomes 1, — 1z7,+22?,—32r°,+ etc., corresponding to 
the differentials of a negative function: and fivally when the 
unit of operation is circulative, we enter upon the differentiation 
of exponential functions. Nature, in the case of a body falling 
in a vacuum under the influence of the force of gravitation, actu- 
ally develops the function (z-+/)?; and it is the inteution of the 
Calculus of Operations to exhibit the harmony which subsists 
between the objective and subjective, between the phenomena 
of the external world, and the mental processes which investigate 
and record them. 
Arr. XXXVL—On the Mammoth Cave of Kentucky ; by Prof. 
B. Situman, Jr. m a letter addressed to Prof. Guyot, 
Cambridge; dated Louisville, November 8, 1850.*) 
I wave lately had an opportunity of visiting the Mammoth 
Cave of Kentucky and have made in connection with Mr. R. N, 
Mantell a collection of the animals found there, which we pro- 
pose to send to our much valued friend, Prof. Agassiz, by an 
early opportunity. My object in writing to you is to say a wort 
regarding the general topographical features of the country and 
some other matters which I suppose will interest you. 
At the level of the Ohio below the falls on which the eity 
stands, are found the lower beds. of silurian rocks and upper 
Devonian, the oldest palzeozoic rocks to be seen in all this region. 
This city is placed on a plateau elevated some seventy-five feet | 
above the low-water mark of the Ohio, and the plain on which 
it stands represents the general level of the surrounding country 
for a long distance above and below. Mammoth Cave lies in @ 
direction nearly south of Louisville on the Green river ‘apd on 
the direct road to Nashville, midway between Louisville and the 
latter city. Leaving Louisville, the stage road follows the main 
course of the Ohio and generally along its banks to the Salt river, 
some twenty miles ; crossing this river the road bears south, leav- 
ing the Ohio on the right, and soon enters a new region. a 
gentle and easy ascent you continue to rise for several miles by.@ 
+ eircuitous path, passing in succession all the members of the sl- 
_[* The following letter was written with no intention of its being published and 
only.for the gratification of a friend, who has sin¢e urged its publication. It is 
therefore given exactly, as it was first written, which may, I trust, be deemed a sufi- 
cient apology for whatever want of fullness of detail attaches toa general and 
sketch. —B. 8., Jr.] Bee pee 
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