ELISHA MITCHELL SCIENTIFIC SOCIETY. 9 



of measure, we are confronted with the difficulty, in the 

 case of any curve line, that we cannot apply the unit of 

 measure, or any fractional part thereof, to the curve. We 

 can apply it, however, to the inscribed or circumscribed 

 polygon, and by taking the limit to which these polygons 

 approach indefinitely as the number of sides is increased 

 indefinitely, we get what is called the length of the curve. 

 Similarly no meaning can be attached to the expressions, 

 area of a curve or area of a curved surface, unless we define 

 them as the limit of the area of the inscribed or circum- 

 scribed polygon in the first case, or as the limit of the area 

 of the surface of the inscribed or circumscribed polyedrons 

 in the second case, the number of sides or faces, as the case 

 may be, increasing indefinitely. In the case of volumes, 

 too, neither the unit of measure nor any fraction of it can 

 be directly applied when the bounding surfaces are curved, 

 so that a volume must be defined as the limit of the varia- 

 ble volume of some inscribed or circumscribed polvedron 

 as the number of faces is indefinitely increased. The dif- 

 ficulty of measuring curved surfaces, volumes, etc., occurs 

 to every reflecting student, and it is strange that none of 

 our geometries give an\' definitions but only methods of 

 finding lengths, areas and volumes of curved lines, surfaces 

 and volumes, assuming that the student will find out in 

 some way what is meant by such terms. 



The "Theory of Limits" will not be entered into here 

 as it is sufficiently exposed in many text-books. Some 

 strange definitions of infinity, though, appear in some 

 excellent books. The following is a sample: "When a 

 variable is conceived to have a value greater than any 

 assigned value, however great this assigned value may be, 

 the variable is said to become infinite ; such a variable is 

 called an infinite mtniber.'' As an "assigned value" 

 means some finite value, it follows from this definition that 

 an infinite number is only some number greater than some 



