We have before seen that not even in this case can we have a lineal value strictly infinite; because this right line only forms a part of the sum of lines which we can imagine. Is it then infinite, or is it not? It is both, if we make the proper distinction. It will be infinite, or we shall have the idea of infinity or negation of limit, applied to a right line alone; but if instead of one right line alone, we take a lineal value, without any condition, the supposed line will not be infinite; the negation of the limit is not applied under that condition; the result must therefore be different.
63. We find the same anomaly, if we take two lines alone. Let us suppose a right line infinitely produced in both directions, and by its side let us describe a curve with continual undulations extending infinitely in a direction parallel to the right line. Both lines will be infinite if we consider only their direction, abstracting their lineal value; but if we regard this value the curve is greater than the straight line; for it is evident if we take a part of the curve corresponding to a part of the straight line, and extend or straighten this part of the curve, it will be greater than the corresponding part of the straight line; as this may be done throughout the whole length of the lines, the lineal value of the curve must be greater than that of the straight line in proportion to the law of its undulations.
64. This may suffice to show how the idea of infinity may be applied under different conditions and produce different results, without any contradiction. What is infinite under one aspect is not so under another aspect; hence we have the orders of infinities which figure so largely in mathematics; but I say again that these contradictions are not susceptible of any explanation if we attribute an absolute value to the idea of the infinite, instead of considering it as the abstract representation of the negation of limit.
65. Is it possible to conceive in a right line or curve an absolutely infinite length or lineal value, to which we may apply the negation of limit absolutely? I think not: for whatever be the line under consideration we can always draw others, which, added to the first, will give a value greater than that of the first above. This is a case in which there is a contradiction between the negation of limit and the condition to which it is subjected. You demand a lineal value to which the negation of limit may be applied absolutely; and on the other hand you require that this lineal value should be found in a determinate line, which by the fact of its being determinate, excludes the absolute negation of limit. The problem supposes contradictory data; therefore the result must be a contradiction.
66. What must we suppose in order to conceive an absolutely infinite lineal value? We need only suppose no condition which excludes the absolute negation of limit. We must here distinguish between the pure conception and the sensible intuition in which it is expressed. The conception of infinite lineal value exists from the moment that we unite the two general conceptions of lineal value and negation of limit. But the sensible intuition, which may represent this conception, is not so easy to imagine, even in general. To arrive at it we must imagine a space without any limit; and then considering in general all the lines whether right lines or curves, which may be drawn in it, in all directions, and under all possible conditions, we must take the sum of all these lineal values; and the result will be an absolutely infinite lineal value; for we shall have applied the negation of limit without any restriction.
67. We may obtain in the same way an infinite superficial value; for it is evident that we may apply to it all that we have said of lineal values.
68. In all these cases we apply the negation of limit to extension considered only in some of its dimensions. If we wish to obtain an absolutely infinite extension, we must abstract no dimension; consequently the absolutely infinite of this order, is extension in all its dimensions with the absolute negation of limit. But it is also to be observed that we must presuppose an absolutely infinite value of extension in order to obtain an absolutely infinite value of lines or surfaces; because it is equivalent to presupposing an infinite space in which the lines and surfaces may be drawn in all directions and under all possible conditions.
[CHAPTER IX.]
CONCEPTION OF AN INFINITE NUMBER.