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If we suppose it produced in the direction of the south, it will be greater by how much it is produced towards the south ; and if it be infin- itely produced in this direction, its length will be twice that of the first line. 34 By the infinite prolongation of a line in two opposite directions we seem to obtain an absolutely infinite line ; for we cannot conceive a lineal value greater than that of a right line infinitely prolonged in opposite directions. But it is not so : by the side of this right line another may be drawn, either finite or infinite, and the sum of the two will form a lineal value greater than that of the first line ; there- fore that line is not infinite, because it is possible to find another still greater. And as, on the other hand, we may draw infinite lines and prolong them infinitely, it follows that none of them can form an infinite lineal value, because it is only a part of the lineal sum resulting from tke addi- tion of all the lines. 35 Reflecting on this apparent contradiction in our ideas, we discover that the idea of the infinite is indetermi- nate, and consequently susceptible of different applications. prev next |