Jnjinite Divisibility of Fmilt Jlalter. 3bl 



will draw a line from A ^Jl B, and, if you will ntiove your 



penknife first over half of it from A lo B, and then over half 

 the remainder, and again over half the remainder, and in the 

 same ratio keep moving your knife, as fast as you can, and 

 you can never move it from A to B Experience proves that 

 one can move his knife from A to B ; and what experience 

 proves is always true : therefore the proposition involves an 

 impossibility. The proposition is predicated on the opinion, 

 that every finite line contains an infinite number of parts of 

 smaller lines, which can be divided ad infinitum But it is im- 

 possible that a finite line can contain an infinite number of 

 parts, or smaller lines ; for any number of equal parts into 

 which every finite line can be divided, will be a definite 

 number of parts, of equal lengths ; and all the parts being 

 equal to the whole, if you keep constantly taking ihe parts of 

 a finite quantity, you will, if you take fast enough, eventually 

 take the whole. If you first take half the parts, and then half 

 the remainder, and again half of the remainder, and so on, ac- 

 cording to this ratio, you will eventually have but one part 

 left, out of any definite number of parts whatever. If the 

 line be divided into parts as small as they possibly can be, 

 so small that no one of them can possibly be any smaller; if 

 you have moved the knife to within the least possible part of 

 the end of the line, or over all the definite parts of it but the 

 very last one; if this last remainder be the least possible part 

 of the line, how can it be divided into less remainders.'' If it 

 is now the least possible part, how can it be divided into less 

 parts ? Suppose you put the point of your knife on the end 

 of the line at B, and move it the least possible part of the line 

 towards A, is it not clearly perceived by the mind, that the part 

 moved over is as small as any part possibly can be ; that it 

 cannot contain parts smaller than itsell ? for no part can be 

 less than the least possible; therefore the least possible part 

 is indivisible. For this reason, when there is but one part of 

 the line left, and that the least possible, if you move the knife 

 any further, you must move it to the end of the line. It is 

 asked, if the least possible part of the line has not a beginning, 

 a middle, and an end ? Every indivisible part has a begin- 

 ning and an end ; but there is no absolute distance between 

 them, though the beginning and the end together make abso- 

 lute distance. The least possible part is so small that it can- 

 not be divided into halves, each of which shall have a begin- 

 ning, a middle, and an end, and an infinite number of smaller 



