Holmes's Notes on the Geology of Charleston. 187 



theory of numbers would indicate. However many divisions into 

 aliquot parts may then be effected, there always must be positions 

 on the line, included between the points of division so obtained ; 

 since no two of the successive divisions of the whole line can 

 agree, as has already been shown, at all points. In other words, 

 there must exist positions between the points of division so ob- 

 tained, which no division of the line into fractions, (i. e., aliquot 

 parts or their aggregates,) however numerous, can ever mark. At 

 any or all such positions, the line would be divided into two 

 parts incommensurable with the whole, and of course incommen- 

 surable with each other. 



The combination, by addition, of the original line or unit and 

 a line equivalent to such a portion, would be of a length which 

 may be represented as between 1 and 2 such units, but the excess 

 above 1, such as cannot be expressed by any fraction, &c, &c. 



The like principles must be applicable to the case of any 

 other quantity which will admit of the like successive fractional 

 division. 



The fundamental reason for the existence of incommensurable 

 quantities seems, then, to be, more concisely, this : The division 

 into fractions is a division into aliquot portions, or implies such a 

 division of the whole as the aggregate of such portions would 

 furnish. Now this is so far from being the only mode of origin- 

 ally dividing the quantity, that it must be regarded as a peculiar 

 and restricted one ; in so much that it would almost seem that the 

 result of a fractional division is not that which would most prob- 

 ably be obtained, if the quantity were divided at hazard; or the 

 chances would be more numerous, that the quantity would be 

 divided incommensurably, than that it would be divided into frac- 

 tions properly so called. 



(To be continued.) 



Akt. XVI.— Notes on the Geology of Charleston, 8. C. ; by 

 " S. Holmes, Corresponding Member of the Acad, of Nat. 

 Sciences, Philadelphia. 



Th at Charleston, the Capital of South Carolina, is built upon 

 geological formations identical in age, and in other respects mi- 

 J a r to those upon which the gnat cities of London and Paris are 

 located, n a curious fact but lately ascertained. The basin shaped 

 depression of its underlying calcareous and other beds, as deter- 

 mined in the survey just made by Proff sor Tuomey, occupies a 

 considerable extent between the Savannah and Peedce Rivers, 

 ar >d rests upon an older group of rocks known to geologists as 

 the Cretaceous formation. The sides of this basin are estimated 

 to be of sufficient inclination to produce those artificial fountains, 





