be made to spell out any words, and with two lamps, distinguished by colour, position, or any other circumstance, we could at once represent Bacon’s biliteral alphabet. Babbage ingeniously suggested that every lighthouse in the world should be made to spell out its own name or number perpetually, by flashes or obscurations of various duration and succession. A system like that of Babbage is now being applied to lighthouses in the United Kingdom by Sir W. Thomson and Dr. John Hopkinson.
Let us calculate the numbers of combinations of different orders which may arise out of the presence or absence of a single mark, say A. In these figures
A A A A
A A A A A A A A
we have four distinct varieties. Form them into a group of a higher order, and consider in how many ways we may vary that group by omitting one or more of the component parts. Now, as there are four parts, and any one may be present or absent, the possible varieties will be 2 × 2 × 2 × 2, or 16 in number. Form these into a new whole, and proceed again to create variety by omitting any one or more of the sixteen. The number of possible changes will now be 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2, or 216, and we can repeat the process again and again. We are imagining the creation of objects, whose numbers are represented by the successive orders of the powers of two.
At the first step we have 2; at the next 22, or 4; at the third (22)2, or 16, numbers of very moderate amount. Let the reader calculate the next term, ((22)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,