III. — On the Continuity of Chance. 

 By ELLERY W. DAVIS. 



Need of a solid and unyielding foundation for the structure 

 that the mathematicians of the day are rearing has caused a 

 thorough overhauling of even the most elementary notions. 

 Nothing escapes this penetrating criticism. Even supposed 

 axioms are contradicted. 



It is proved, for instance, that, after all, two parallel lines may 

 meet if sufficiently produced; that through a given point many 

 distinct parallels may be drawn to one and the same straight 

 line; that a part may be equal to the whole; that to multiply 

 the first of two quantities by the second may give a totally dif- 

 ferent result from that gotten by multiplying the second by 

 the first; that lines may be perpendicular to themselves; that a 

 closed hollow shell may be turned inside out without straining 

 or breaking it. 



Among the notions thus carefully examined are those of con 

 tinuity and discontinuity. Of the one, a line is a simple ex- 

 ample; of the other, a row of points. Nothing, at first blush, 

 seems plainer than the distinction between them. It is when 

 we come to consider discontinuity that almost merges into con- 

 tinuity that difficulty of definition presents itself Similar, I 

 take it, is the difficulty a biologist has in distinguishing be- 

 tween plant and animal among the lower organisms. Let us 

 study the solution of the mathematical problem. 



Take, for a sample of continuity, an inch-long line. Divide 

 the line into ten equal parts, marking the points of division; 

 repeat this operation upon each of the parts, upon the parts of 

 the parts. Imagine the operation continued indefinitely. Then 

 there is marked upon the line every distance from either end 

 that can be expressed by a decimal of an inch. To fix our 

 ideas, suppose the measurement to be always from the left- 



Umveusitv Stldies, Vol. II., No. 2, January, 1S97. 131 



