37. What we have observed of lineal values is also true of numerical values expressed in series. Mathematics speak of infinite series, but there can be no such series. Let the series be _a_, _b_, _c_, _d_, _e_, ....: it is called infinite if its terms continue _ad infinitum_. It cannot be denied that the series is infinite under one aspect; for there is no limit which puts an end to it in one sense; but it is evident that the number of its terms will never be infinite, because there are others greater; such, for instance, is the series continued from left to right, if continued from right to left at the same time, in this manner: .......... _e_, _d_, _c_, _b_, | _a_, _b_, _c_, _d_, _e_, .......... In this case the number of terms is evidently twice as great as in the first series. Therefore the series which are called infinite are not infinite, and cannot be so, in the strict sense of the term. 38. But what is still more strange is, that the series is not infinite, even though we suppose it continued in opposite directions; for by its side we may imagine another, and the sum of the terms of both will be greater than the terms of either; therefore neither will be infinite. As it is evident that whatever be the series, we can always imagine others, it follows that there can be no infinite series in the sense in which mathematicians use the word series to express a continuation of terms, not excluding the possibility of other continuations besides the supposed infinite continuation. 39. The objections against lineal infinity apply equally to surfaces. If we suppose an infinite plane, it is evident that we can describe an infinity of planes distinct from the first plain and intersecting it in a variety of angles; the sum of all these surfaces will be greater than any one of them. Therefore the infinite extension of a plain in all directions does not constitute a truly infinite surface. 40. A solid expanding in all directions seems to be infinite; but if we consider that the mathematical idea of a solid does not involve impenetrability, we shall see that inside of the first solid a second may be placed, which, added to the first, will give a value double that of the first alone. Let S be the empty space which we imagine to be infinite; and let W be a world of equal extension placed in it and filling it; it is evident that S + W are greater than S alone. Therefore, although we suppose S to be infinite, = ∞, W also = ∞; therefore S + W = ∞ + ∞ = 2 ∞. And as this value expresses the size, the first is not infinite because it can be doubled. If we take the impenetrability, the operation may proceed _ad infinitum_. Therefore the first infinite, far from being infinite, seems to be a quantity susceptible of infinite increase. CHAPTER VI. ORIGIN OF THE VAGUENESS AND APPARENT CONTRADICTIONS IN THE APPLICATION OF THE IDEA OF THE INFINITE.