194 THE PRINCIPLES OF SCIENCE. [CHAP. 



short, may 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 dif- 

 ferent orders which may arise out of the presence or 

 absence of a single mark, say A. In these figures 



LAJ _ I 



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 x 2 x 2 X 2, or 1 6 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 pos- 

 sible changes will now be 2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2, or 

 2 16 , 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 2 2 , or 4 ; 



at the third 2 2 , or 16, numbers of very moderate amount. 



2 2 



Let the reader calculate the next term, 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 



