But this Infinitesimal area, inconceivably small as it is, is nevertheless greater than Zero. Hence our continuously-diminishing Strip is bound, before reaching Zero, to pass through this singularly unassuming value. And, at that moment, what will be its width? Its length will be Infinite, as usual: its area will be an Infinitesimal of the first order: but its width cannot be an Infinitesimal of the same order as the previous width; for that would yield a finite area, as we have seen. What else, then, can it be but an Infinitesimal of the second order? A Line of such stupendous brevity that no finite multiple of it can even make up an Infinitesimal of the first order. Evidently we might repeat this process ad libitum, and so get Infinitesimals of the third order, the fourth order, and so on. To sum up our results, so far. We see that a Line may be either Finite, or may extend to an Infinity of the first order, or may dwindle to an Infinitesimal of the first, second, or any order we choose: and that an Area may be either Finite, or may extend to an Infinity of the first order (an Infinite-Strip), or of the second order (e.g. the whole Infinite-Plane), or may dwindle to an Infinitesimal of any order we choose. Now we may reasonably expect to find that all, that has been here said, is equally applicable to any kind of Magnitude that is capable of continuous increase and decrease. Let us consider, then, whether it is possible to have Infinitesimal Angles of the various orders. § 3. [Pg 48] Infinitesimal Angles and Sectors. For this purpose, let us return to our upright Infinite-Plane, and, taking some Point at random as a centre, let us imagine two Lines radiating from it (say at an angle of 45°), and both of them extended to infinity, and therefore having between them a Sector, finite in angular magnitude, infinite in length, and therefore infinite in area. I named 45° as my specimen-angle, because it is the one single angle, other than a right angle, with which Society is acquainted. Enquire of some chatty traveller, who is relating his experiences of an Alpine Pass, what slope it was that he had to climb. "The ground sloped at an angle of forty-five," he is sure to reply. Nay, I once met a gentleman who, on hearing it mentioned that the 'dip' of a certain river-bed was, in one place, "one in forty-five," cautiously remarked "I suppose that means an angle of forty-five degrees?" Imagine a river sloping at that angle! And then imagine the labour of rowing up it, and the headlong, wild delight of rowing down it! But this is a digression. Now it clearly is possible, this time, by multiplying this Infinite-Sector by a certain finite number, namely '8,' to make it equal to the whole Infinite-Plane. Hence this Infinite-Sector is an Infinity of the same order as the whole Infinite-Plane; i.e. it is of the second order.