At the first step we have 2; at the next 2^{2}, or 4; at the third (2^{2})^{2}, or 16, numbers of very moderate amount. Let the reader calculate the next term, ((2^{2})^{2})^{2}, and he will be surprised to find it leap up to 65,536. But at the next step he has to calculate the value of 65,536 *two*’s multiplied together, and it is so great that we could not possibly compute it, the mere expression of the result requiring 19,729 places of figures. But go one step more and we pass the bounds of all reason. The sixth order of the powers of *two* becomes so great, that we could not even express the number of figures required in writing it down, without using about 19,729 figures for the purpose. The successive orders of the powers of two have then the following values so far as we can succeed in describing them:-- First order 2 Second order 4 Third order 16 Fourth order 65,536 Fifth order, number expressed by 19,729 figures. Sixth order, number expressed by figures, to express the number of which figures would require about 19,729 figures. It may give us some notion of infinity to remember that at this sixth step`, having long surpassed all bounds of intuitive conception, we make no approach to a limit. Nay, were we to make a hundred such steps, we should be as far away as ever from actual infinity. It is well worth observing that our powers of expression rapidly overcome the possible multitude of finite objects which may exist in any assignable space. Archimedes showed long ago, in one of the most remarkable writings of antiquity, the *Liber de Arcnæ Numero*, that the grains of sand in the world could be numbered, or rather, that if numbered, the result could readily be expressed in arithmetical notation. Let us extend his problem, and ascertain whether we could express the number of atoms which could exist in the visible universe. The most distant stars which can now be seen by telescopes--those of the sixteenth magnitude--are supposed to have a distance of about 33,900,000,000,000,000 miles. Sir W. Thomson has shown reasons for supposing that there do not exist more than from 3 × 10^{24} to 10^{26} molecules in a cubic centimetre of a solid or liquid substance.[109] Assuming these data to be true, for the sake of argument, a simple calculation enables us to show that the almost inconceivably vast sphere of our stellar system if entirely filled with solid matter, would not contain more than about 68 × 10^{90} atoms, that is to say, a number requiring for its expression 92 places of figures. Now, this number would be immensely less than the fifth order of the powers of two. [109] *Nature*, vol. i. p. 553.